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  1979  sbrbif  1981  abeq2  2305  abid2f  2365  necon4biddc  2442  pm13.183  2902  disj3  3504  euabsn2  3692  a9evsep  4156  inex1  4168  zfpair2  4244  sucel  4446  bdinex1  15629  bj-zfpair2  15640  bj-d0clsepcl  15655