Theorem bibi12i 340

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 338	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃))
3		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	bibi1i 339	. 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃))
5	2, 4	bitri 276	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Syntax hints:   ↔ wb 207

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 208

This theorem is referenced by:  pm5.32  578  biadan  824  orbidi  960  pm5.7  961  xorbi12i  1531  norass  1544  rexprg  4629  brsymdif  5131  nfnid  5304  asymref  6066  isocnv2  7275  zfcndrep  10528  f1omvdco3  19415  brtxpsd  36120  eliminable-abeqab  37221  bj-sbeq  37254  bj-rcleqf  37378  symrefref3  39015  eldisjn0el  39276  abbibw  43127  rp-fakeoranass  43958  rp-fakeinunass  43959  relexp0eq  44145  permaxext  45449  absnsb  47490  ichcom  47934  ichbi12i  47935