Theorem iotabidv 5307

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 5294	. 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 5282

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 3892  df-iota 5284

This theorem is referenced by:  csbiotag  5317  dffv3g  5631  fveq1  5634  fveq2  5635  fvres  5659  csbfv12g  5675  fvco2  5711  riotaeqdv  5967  riotabidv  5968  riotabidva  5984  ovtposg  6420  shftval  11379  sumeq1  11909  sumeq2  11913  zsumdc  11938  isumclim3  11977  isumshft  12044  prodeq1f  12106  prodeq2w  12110  prodeq2  12111  zproddc  12133  pcval  12862  grpidvalg  13449  grpidpropdg  13450  igsumvalx  13465  gsumpropd  13468  gsumpropd2  13469  gsumress  13471  gsumval2  13473  dfur2g  13968  oppr0g  14087  oppr1g  14088  gfsumval  16630