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  954  bimsc1  965  eujust  2044  euf  2047  ceqex  2888  reu6i  2952  axsep2  4149  zfauscl  4150  copsexg  4274  euotd  4284  cnveq0  5123  iotaval  5227  iota5  5237  eufnfv  5790  isoeq1  5845  isoeq3  5847  isores2  5857  isores3  5859  isotr  5860  isoini2  5863  riota5f  5899  caovordg  6088  caovord  6092  dfoprab4f  6248  frecabcl  6454  nnaword  6566  xpf1o  6902  ltanqg  7462  ltmnqg  7463  ltasrg  7832  axpre-ltadd  7948  prmdvdsexp  12289  subrgsubm  13733  bdsep2  15448  bdzfauscl  15452