Theorem bibi12i 341

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 339	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃))
3		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	bibi1i 340	. 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃))
5	2, 4	bitri 277	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Syntax hints:   ↔ wb 208

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 209

This theorem is referenced by:  pm5.32  581  biadan  828  orbidi  965  pm5.7  966  xorbi12i  1544  norass  1557  rexprg  4656  brsymdif  5159  nfnid  5332  asymref  6103  isocnv2  7315  zfcndrep  10572  f1omvdco3  19489  brtxpsd  36239  eliminable-abeqab  37350  bj-sbeq  37383  bj-rcleqf  37507  symrefref3  39144  eldisjn0el  39405  abbibw  43256  rp-fakeoranass  44087  rp-fakeinunass  44088  relexp0eq  44274  permaxext  45578  absnsb  47618  ichcom  48062  ichbi12i  48063