Theorem iotabidv 5360

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 5347	. 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 5335

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 5337

This theorem is used by:  csbiotag  5370  dffv3g  5691  fveq1  5694  fveq2  5695  fvres  5719  csbfv12g  5736  fvco2  5774  riotaeqdv  6039  riotabidv  6040  riotabidva  6056  ovtposg  6530  shftval  11592  sumeq1  12123  sumeq2  12127  zsumdc  12153  isumclim3  12192  isumshft  12259  prodeq1f  12321  prodeq2w  12325  prodeq2  12326  zproddc  12348  pcval  13077  grpidvalg  13695  grpidpropdg  13696  gzsumvalx  13711  gzsumress  13714  gzsumval2  13716  gsumvalfi  14154  dfur2g  14268  oppr0g  14389  oppr1g  14390