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  820  nanass  1510  xorass  1515  hadbi  1598  hadcoma  1599  hadnot  1602  sbrbis  2310  csbied  3910  dfss2  3944  ssequn1  4161  ab0w  4354  asymref  6105  aceq1  10131  aceq0  10132  zfac  10474  zfcndac  10633  hashreprin  34652  axacprim  35724  eliminable-abeqv  36885  wl-3xorcoma  37496  wl-3xornot  37499  redundpbi1  38649  onsupmaxb  43263  rp-fakeanorass  43537  ichn  47470  dfich2  47472