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  2047  euf  2050  ceqex  2891  reu6i  2955  axsep2  4153  zfauscl  4154  copsexg  4278  euotd  4288  cnveq0  5127  iotaval  5231  iota5  5241  eufnfv  5796  isoeq1  5851  isoeq3  5853  isores2  5863  isores3  5865  isotr  5866  isoini2  5869  riota5f  5905  caovordg  6095  caovord  6099  dfoprab4f  6260  frecabcl  6466  nnaword  6578  xpf1o  6914  ltanqg  7484  ltmnqg  7485  ltasrg  7854  axpre-ltadd  7970  prmdvdsexp  12341  subrgsubm  13866  bdsep2  15616  bdzfauscl  15620