Theorem iotabidv 5259

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 1898	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 5246	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 14	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 105  ∀wal 1371   = wceq 1373  ℩cio 5235

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-uni 3853  df-iota 5237

This theorem is referenced by:  csbiotag  5269  dffv3g  5579  fveq1  5582  fveq2  5583  fvres  5607  csbfv12g  5621  fvco2  5655  riotaeqdv  5907  riotabidv  5908  riotabidva  5923  ovtposg  6352  shftval  11180  sumeq1  11710  sumeq2  11714  zsumdc  11739  isumclim3  11778  isumshft  11845  prodeq1f  11907  prodeq2w  11911  prodeq2  11912  zproddc  11934  pcval  12663  grpidvalg  13249  grpidpropdg  13250  igsumvalx  13265  gsumpropd  13268  gsumpropd2  13269  gsumress  13271  gsumval2  13273  dfur2g  13768  oppr0g  13887  oppr1g  13888