Theorem iotabidv 5238

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 5225	. 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 5214

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 3837  df-iota 5216

This theorem is referenced by:  csbiotag  5248  dffv3g  5551  fveq1  5554  fveq2  5555  fvres  5579  csbfv12g  5593  fvco2  5627  riotaeqdv  5875  riotabidv  5876  riotabidva  5891  ovtposg  6314  shftval  10972  sumeq1  11501  sumeq2  11505  zsumdc  11530  isumclim3  11569  isumshft  11636  prodeq1f  11698  prodeq2w  11702  prodeq2  11703  zproddc  11725  pcval  12437  grpidvalg  12959  grpidpropdg  12960  igsumvalx  12975  gsumpropd  12978  gsumpropd2  12979  gsumress  12981  gsumval2  12983  dfur2g  13461  oppr0g  13580  oppr1g  13581