Theorem iotabidv 5173

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 1862	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5161	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1341   = wceq 1343  ℩cio 5150

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-rex 2449  df-uni 3789  df-iota 5152

This theorem is referenced by:  csbiotag  5180  dffv3g  5481  fveq1  5484  fveq2  5485  fvres  5509  csbfv12g  5521  fvco2  5554  riotaeqdv  5798  riotabidv  5799  riotabidva  5813  ovtposg  6223  shftval  10763  sumeq1  11292  sumeq2  11296  zsumdc  11321  isumclim3  11360  isumshft  11427  prodeq1f  11489  prodeq2w  11493  prodeq2  11494  zproddc  11516  pcval  12224