Theorem bibi12i 339

Description: The equivalence of two equivalences. (Contributed by NM, 26-May-1993.)

Hypotheses

Ref	Expression
bibi2i.1	⊢ (𝜑 ↔ 𝜓)
bibi12i.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
bibi12i	⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Proof of Theorem bibi12i

Step	Hyp	Ref	Expression
1		bibi12i.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
2	1	bibi2i 337	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃))
3		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	bibi1i 338	. 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃))
5	2, 4	bitri 275	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Syntax hints:   ↔ wb 206

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This theorem depends on definitions:  df-bi 207

This theorem is referenced by:  pm5.32  573  biadan  818  orbidi  954  pm5.7  955  xorbi12i  1525  norass  1538  rexprg  4654  brsymdif  5157  nfnid  5320  asymref  6073  isocnv2  7277  zfcndrep  10525  f1omvdco3  19378  brtxpsd  36086  eliminable-abeqab  37069  bj-sbeq  37102  bj-rcleqf  37226  symrefref3  38821  eldisjn0el  39065  abbibw  42920  rp-fakeoranass  43755  rp-fakeinunass  43756  relexp0eq  43942  permaxext  45246  absnsb  47273  ichcom  47705  ichbi12i  47706