Theorem bibi2d 232

Description: Deduction adding a biconditional to the left in an equivalence. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Wolf Lammen, 19-May-2013.)

Hypothesis

Ref	Expression
imbid.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bibi2d	⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒)))

Proof of Theorem bibi2d

Step	Hyp	Ref	Expression
1		imbid.1	. . . . 5 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	pm5.74i 180	. . . 4 ⊢ ((𝜑 → 𝜓) ↔ (𝜑 → 𝜒))
3	2	bibi2i 227	. . 3 ⊢ (((𝜑 → 𝜃) ↔ (𝜑 → 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒)))
4		pm5.74 179	. . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜓)))
5		pm5.74 179	. . 3 ⊢ ((𝜑 → (𝜃 ↔ 𝜒)) ↔ ((𝜑 → 𝜃) ↔ (𝜑 → 𝜒)))
6	3, 4, 5	3bitr4i 212	. 2 ⊢ ((𝜑 → (𝜃 ↔ 𝜓)) ↔ (𝜑 → (𝜃 ↔ 𝜒)))
7	6	pm5.74ri 181	1 ⊢ (𝜑 → ((𝜃 ↔ 𝜓) ↔ (𝜃 ↔ 𝜒)))

Syntax hints:   → wi 4   ↔ 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:  bibi1d  233  bibi12d  235  biantr  958  bimsc1  969  eujust  2079  euf  2082  ceqex  2931  reu6i  2995  axsep2  4206  zfauscl  4207  copsexg  4334  euotd  4345  cnveq0  5191  iotaval  5296  iota5  5306  eufnfv  5880  isoeq1  5937  isoeq3  5939  isores2  5949  isores3  5951  isotr  5952  isoini2  5955  riota5f  5993  caovordg  6185  caovord  6189  dfoprab4f  6351  frecabcl  6560  nnaword  6674  xpf1o  7025  ltanqg  7613  ltmnqg  7614  ltasrg  7983  axpre-ltadd  8099  prmdvdsexp  12713  subrgsubm  14241  wlkeq  16165  bdsep2  16431  bdzfauscl  16435