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  390  sblbis  2011  sbrbif  2013  abeq2  2338  abid2f  2398  necon4biddc  2475  pm13.183  2941  disj3  3544  euabsn2  3735  a9evsep  4205  inex1  4217  zfpair2  4293  sucel  4498  bdinex1  16192  bj-zfpair2  16203  bj-d0clsepcl  16218