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  4658  brsymdif  5164  nfnid  5337  asymref  6110  isocnv2  7337  zfcndrep  10692  f1omvdco3  19656  angmgmaddcpbl  29383  brtxpsd  36636  eliminable-abeqab  37760  bj-sbeq  37793  bj-rcleqf  37918  bj-vn0ALT  37967  symrefref3  39560  eldisjn0el  39821  abbibw  43668  rp-fakeoranass  44499  rp-fakeinunass  44500  relexp0eq  44686  permaxext  45973  absnsb  48066  ichcom  48510  ichbi12i  48511