Theorem bibi1i 341

Description: Inference adding a biconditional to the right in an equivalence. (Contributed by NM, 26-May-1993.)

Hypothesis

Ref	Expression
bibi2i.1	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bibi1i	⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))

Proof of Theorem bibi1i

Step	Hyp	Ref	Expression
1		bicom 225	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜒 ↔ 𝜑))
2		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	bibi2i 340	. 2 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))
4		bicom 225	. 2 ⊢ ((𝜒 ↔ 𝜓) ↔ (𝜓 ↔ 𝜒))
5	1, 3, 4	3bitri 300	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))

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:  bibi12i  342  biluk  389  biadaniALT  832  nanass  1533  xorass  1538  hadbi  1621  hadcoma  1622  hadnot  1625  sbrbis  2346  csbied  3891  dfss2  3925  ssequn1  4141  asymref  6106  aceq1  10089  aceq0  10090  zfac  10432  zfcndac  10592  hashreprin  34919  axacprim  36065  eliminable-abeqv  37359  wl-3xorcoma  37979  wl-3xornot  37982  redundpbi1  39221  onsupmaxb  43823  rp-fakeanorass  44096  ichn  48061  dfich2  48063