Theorem iotabidv 5363

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 1927	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5350	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

This proof depends on syntax axioms:   → wi 4   ↔ wb 105  ∀wal 1400   = wceq 1402  ℩cio 5338

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-uni 3936  df-iota 5340

This theorem is used by:  csbiotag  5373  dffv3g  5694  fveq1  5697  fveq2  5698  fvres  5722  csbfv12g  5739  fvco2  5777  riotaeqdv  6042  riotabidv  6043  riotabidva  6059  ovtposg  6534  shftval  11613  sumeq1  12147  sumeq2  12151  zsumdc  12177  isumclim3  12216  isumshft  12283  prodeq1f  12345  prodeq2w  12349  prodeq2  12350  zproddc  12372  pcval  13105  grpidvalg  13753  grpidpropdg  13754  gzsumvalx  13769  gzsumress  13772  gzsumval2  13774  gsumvalfi  14243  dfur2g  14374  oppr0g  14497  oppr1g  14498