Theorem lbeui 99

Description: Linear biconditional is Euclidean. Inference for lbeu 126.

Hypotheses

Ref	Expression
lbeui.1	⊦ (𝜑 ⧟ 𝜓)
lbeui.2	⊦ (𝜑 ⧟ 𝜒)

Assertion

Ref	Expression
lbeui	⊦ (𝜓 ⧟ 𝜒)

Proof of Theorem lbeui

Step	Hyp	Ref	Expression
1		lbeui.1	. . . 4 ⊦ (𝜑 ⧟ 𝜓)
2	1	lbi2 90	. . 3 ⊦ (𝜓 ⊸ 𝜑)
3		lbeui.2	. . . 4 ⊦ (𝜑 ⧟ 𝜒)
4	3	lbi1 89	. . 3 ⊦ (𝜑 ⊸ 𝜒)
5	2, 4	syl 75	. 2 ⊦ (𝜓 ⊸ 𝜒)
6	3	lbi2 90	. . 3 ⊦ (𝜒 ⊸ 𝜑)
7	1	lbi1 89	. . 3 ⊦ (𝜑 ⊸ 𝜓)
8	6, 7	syl 75	. 2 ⊦ (𝜒 ⊸ 𝜓)
9	5, 8	ilb 96	1 ⊦ (𝜓 ⧟ 𝜒)

Syntax hints:   ⧟ wlb 55

This theorem was proved from axioms:  ax-ibot 4  ax-ebot 5  ax-cut 6  ax-init 7  ax-mdco 8  ax-mdas 9  ax-iac 32  ax-eac1 33  ax-eac2 34

This theorem depends on definitions:  df-lb 56  df-li 62

This theorem is referenced by:  lbsymi  100  lbtri  101