Theorem iotabidv 5334

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 1923	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5321	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1396   = wceq 1398  ℩cio 5309

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-rex 2526  df-uni 3914  df-iota 5311

This theorem is referenced by:  csbiotag  5344  dffv3g  5665  fveq1  5668  fveq2  5669  fvres  5693  csbfv12g  5709  fvco2  5745  riotaeqdv  6003  riotabidv  6004  riotabidva  6020  ovtposg  6489  shftval  11506  sumeq1  12036  sumeq2  12040  zsumdc  12066  isumclim3  12105  isumshft  12172  prodeq1f  12234  prodeq2w  12238  prodeq2  12239  zproddc  12261  pcval  12990  grpidvalg  13578  grpidpropdg  13579  igsumvalx  13594  gsumpropd  13597  gsumpropd2  13598  gsumress  13600  gsumval2  13602  dfur2g  14098  oppr0g  14217  oppr1g  14218  gfsumval  16853