Theorem bibi2i 227

Description: Inference adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 7-May-2011.) (Proof shortened by Wolf Lammen, 16-May-2013.)

Hypothesis

Ref	Expression
bibi.a	⊢ (𝜑 ↔ 𝜓)

Assertion

Ref	Expression
bibi2i	⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))

Proof of Theorem bibi2i

Step	Hyp	Ref	Expression
1		id 19	. . 3 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜑))
2		bibi.a	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	1, 2	bitrdi 196	. 2 ⊢ ((𝜒 ↔ 𝜑) → (𝜒 ↔ 𝜓))
4		id 19	. . 3 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜓))
5	4, 2	bitr4di 198	. 2 ⊢ ((𝜒 ↔ 𝜓) → (𝜒 ↔ 𝜑))
6	3, 5	impbii 126	1 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))

Syntax hints:   ↔ wb 105

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108

This theorem depends on definitions:  df-bi 117

This theorem is referenced by:  bibi1i  228  bibi12i  229  bibi2d  232  pm4.71r  394  sblbis  2020  sbrbif  2022  abeq2  2347  abid2f  2418  necon4biddc  2495  pm13.183  2964  ab0w  3550  disj3  3577  euabsn2  3779  a9evsep  4253  inex1  4265  zfpair2  4345  sucel  4553  uniex2  4579  bdinex1  16908  bj-zfpair2  16919  bj-uniex2  16925  bj-d0clsepcl  16934