Theorem bibi12i 228

Description: The equivalence of two equivalences. (Contributed by NM, 5-Aug-1993.)

Hypotheses

Ref	Expression
bibi.a	⊢ (𝜑 ↔ 𝜓)
bibi12.2	⊢ (𝜒 ↔ 𝜃)

Assertion

Ref	Expression
bibi12i	⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Proof of Theorem bibi12i

Step	Hyp	Ref	Expression
1		bibi12.2	. . 3 ⊢ (𝜒 ↔ 𝜃)
2	1	bibi2i 226	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜑 ↔ 𝜃))
3		bibi.a	. . 3 ⊢ (𝜑 ↔ 𝜓)
4	3	bibi1i 227	. 2 ⊢ ((𝜑 ↔ 𝜃) ↔ (𝜓 ↔ 𝜃))
5	2, 4	bitri 183	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜃))

Syntax hints:   ↔ wb 104

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107

This theorem depends on definitions:  df-bi 116

This theorem is referenced by:  pm5.7dc  944  asymref  4989  rexrnmpt  5628  uzennn  10371