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  1524  norass  1537  vn0  4304  ab0orv  4342  rexprg  4657  brsymdif  5161  nfnid  5325  asymref  6077  isocnv2  7288  zfcndrep  10543  f1omvdco3  19355  brtxpsd  35855  eliminable-abeqab  36829  bj-sbeq  36862  bj-rcleqf  36986  symrefref3  38528  eldisjn0el  38771  abbibw  42638  rp-fakeoranass  43476  rp-fakeinunass  43477  relexp0eq  43663  permaxext  44968  absnsb  47001  ichcom  47433  ichbi12i  47434