Theorem bitrns-P3.33c 302

Description: Equivalence Property: '↔' Transitivity. †

Hypotheses

Ref	Expression
bitrns-P3.33c.1	⊢ (𝛾 → (𝜑 ↔ 𝜓))
bitrns-P3.33c.2	⊢ (𝛾 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bitrns-P3.33c	⊢ (𝛾 → (𝜑 ↔ 𝜒))

Proof of Theorem bitrns-P3.33c

Step	Hyp	Ref	Expression
1		bitrns-P3.33c.1	. . . 4 ⊢ (𝛾 → (𝜑 ↔ 𝜓))
2	1	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜑 → 𝜓))
3		bitrns-P3.33c.2	. . . 4 ⊢ (𝛾 → (𝜓 ↔ 𝜒))
4	3	ndbief-P3.14 179	. . 3 ⊢ (𝛾 → (𝜓 → 𝜒))
5	2, 4	syl-P3.24 259	. 2 ⊢ (𝛾 → (𝜑 → 𝜒))
6	3	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜒 → 𝜓))
7	1	ndbier-P3.15 180	. . 3 ⊢ (𝛾 → (𝜓 → 𝜑))
8	6, 7	syl-P3.24 259	. 2 ⊢ (𝛾 → (𝜒 → 𝜑))
9	5, 8	ndbii-P3.13 178	1 ⊢ (𝛾 → (𝜑 ↔ 𝜒))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133

This theorem is referenced by:  bitrns-P3.33c.RC  303  bitrns-P3.33c.CL  304  subbil-P3.41a-L1  331  subbir-P3.41b  334  subbid-P3.41c  336  subandr-P3.42b  341  subandd-P3.42c  343  suborr-P3.43b  348  subord-P3.43c  350  dbitrns-P4.16  428  tbitrns-P4.17  430  subbir2-P4.RC  549  subeqd-P5  637  subelofd-P5  642  example-E6.01a  706  example-E6.02a  712  psubsuccv-P6-L1  805  psubaddv-P6-L1  807  psubmultv-P6-L1  809  ndsubeqd-P7  856  ndsubelofd-P7  857  psubinv-P7  939