Theorem bibi12i 342

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 340	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃))
3		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	bibi1i 341	. 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃))
5	2, 4	bitri 278	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Syntax hints:   ↔ wb 209

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 210

This theorem is referenced by:  pm5.32  583  biadan  830  orbidi  967  pm5.7  968  xorbi12i  1554  norass  1567  rexprg  4663  brsymdif  5170  nfnid  5346  asymref  6116  isocnv2  7329  zfcndrep  10594  f1omvdco3  19514  brtxpsd  36384  eliminable-abeqab  37503  bj-sbeq  37536  bj-rcleqf  37661  bj-vn0ALT  37708  symrefref3  39297  eldisjn0el  39558  abbibw  43409  rp-fakeoranass  44240  rp-fakeinunass  44241  relexp0eq  44427  permaxext  45714  absnsb  47764  ichcom  48208  ichbi12i  48209