Theorem bibi1i 338

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 222	. 2 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜒 ↔ 𝜑))
2		bibi2i.1	. . 3 ⊢ (𝜑 ↔ 𝜓)
3	2	bibi2i 337	. 2 ⊢ ((𝜒 ↔ 𝜑) ↔ (𝜒 ↔ 𝜓))
4		bicom 222	. 2 ⊢ ((𝜒 ↔ 𝜓) ↔ (𝜓 ↔ 𝜒))
5	1, 3, 4	3bitri 297	1 ⊢ ((𝜑 ↔ 𝜒) ↔ (𝜓 ↔ 𝜒))

Syntax hints:   ↔ wb 206

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 207

This theorem is referenced by:  bibi12i  339  biluk  385  biadaniALT  821  nanass  1512  xorass  1517  hadbi  1600  hadcoma  1601  hadnot  1604  sbrbis  2316  csbied  3874  dfss2  3908  ssequn1  4127  asymref  6073  aceq1  10030  aceq0  10031  zfac  10373  zfcndac  10533  hashreprin  34780  axacprim  35905  eliminable-abeqv  37190  wl-3xorcoma  37808  wl-3xornot  37811  redundpbi1  39050  onsupmaxb  43685  rp-fakeanorass  43958  ichn  47928  dfich2  47930