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))

This proof depends on syntax axioms:   ↔ wb 209

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210

This theorem is used by:  pm5.32  584  biadan  831  orbidi  967  pm5.7  968  xorbi12i  1554  norass  1567  rexprg  4665  brsymdif  5172  nfnid  5348  asymref  6118  isocnv2  7338  zfcndrep  10614  f1omvdco3  19563  brtxpsd  36421  eliminable-abeqab  37560  bj-sbeq  37593  bj-rcleqf  37718  bj-vn0ALT  37765  symrefref3  39355  eldisjn0el  39616  abbibw  43467  rp-fakeoranass  44298  rp-fakeinunass  44299  relexp0eq  44485  permaxext  45772  absnsb  47822  ichcom  48266  ichbi12i  48267