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  1551  norass  1564  rexprg  4668  brsymdif  5174  nfnid  5347  asymref  6117  isocnv2  7330  zfcndrep  10598  f1omvdco3  19518  brtxpsd  36282  eliminable-abeqab  37391  bj-sbeq  37424  bj-rcleqf  37548  bj-vn0ALT  37595  symrefref3  39186  eldisjn0el  39447  abbibw  43300  rp-fakeoranass  44131  rp-fakeinunass  44132  relexp0eq  44318  permaxext  45605  absnsb  47652  ichcom  48096  ichbi12i  48097