Theorem iotabidv 5237

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 1885	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5224	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1362   = wceq 1364  ℩cio 5213

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-uni 3836  df-iota 5215

This theorem is referenced by:  csbiotag  5247  dffv3g  5550  fveq1  5553  fveq2  5554  fvres  5578  csbfv12g  5592  fvco2  5626  riotaeqdv  5874  riotabidv  5875  riotabidva  5890  ovtposg  6312  shftval  10969  sumeq1  11498  sumeq2  11502  zsumdc  11527  isumclim3  11566  isumshft  11633  prodeq1f  11695  prodeq2w  11699  prodeq2  11700  zproddc  11722  pcval  12434  grpidvalg  12956  grpidpropdg  12957  igsumvalx  12972  gsumpropd  12975  gsumpropd2  12976  gsumress  12978  gsumval2  12980  dfur2g  13458  oppr0g  13577  oppr1g  13578