Theorem iotabidv 5304

Description: Formula-building deduction for iota. (Contributed by NM, 20-Aug-2011.)

Hypothesis

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

Assertion

Ref	Expression
iotabidv	⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Distinct variable group:   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem iotabidv

Step	Hyp	Ref	Expression
1		iotabidv.1	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	alrimiv 1920	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5291	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1393   = wceq 1395  ℩cio 5279

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-uni 3889  df-iota 5281

This theorem is referenced by:  csbiotag  5314  dffv3g  5628  fveq1  5631  fveq2  5632  fvres  5656  csbfv12g  5672  fvco2  5708  riotaeqdv  5964  riotabidv  5965  riotabidva  5981  ovtposg  6416  shftval  11357  sumeq1  11887  sumeq2  11891  zsumdc  11916  isumclim3  11955  isumshft  12022  prodeq1f  12084  prodeq2w  12088  prodeq2  12089  zproddc  12111  pcval  12840  grpidvalg  13427  grpidpropdg  13428  igsumvalx  13443  gsumpropd  13446  gsumpropd2  13447  gsumress  13449  gsumval2  13451  dfur2g  13946  oppr0g  14065  oppr1g  14066