Theorem iotabidv 5316

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 1922	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5303	. 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 5291

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 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-rex 2517  df-uni 3899  df-iota 5293

This theorem is referenced by:  csbiotag  5326  dffv3g  5644  fveq1  5647  fveq2  5648  fvres  5672  csbfv12g  5688  fvco2  5724  riotaeqdv  5982  riotabidv  5983  riotabidva  5999  ovtposg  6468  shftval  11446  sumeq1  11976  sumeq2  11980  zsumdc  12006  isumclim3  12045  isumshft  12112  prodeq1f  12174  prodeq2w  12178  prodeq2  12179  zproddc  12201  pcval  12930  grpidvalg  13517  grpidpropdg  13518  igsumvalx  13533  gsumpropd  13536  gsumpropd2  13537  gsumress  13539  gsumval2  13541  dfur2g  14037  oppr0g  14156  oppr1g  14157  gfsumval  16789