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  952  bimsc1  963  eujust  2028  euf  2031  ceqex  2865  reu6i  2929  axsep2  4123  zfauscl  4124  copsexg  4245  euotd  4255  cnveq0  5086  iotaval  5190  iota5  5199  eufnfv  5748  isoeq1  5802  isoeq3  5804  isores2  5814  isores3  5816  isotr  5817  isoini2  5820  riota5f  5855  caovordg  6042  caovord  6046  dfoprab4f  6194  frecabcl  6400  nnaword  6512  xpf1o  6844  ltanqg  7399  ltmnqg  7400  ltasrg  7769  axpre-ltadd  7885  prmdvdsexp  12148  subrgsubm  13355  bdsep2  14641  bdzfauscl  14645