Theorem iotabidv 6470

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 1927	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 ↔ 𝜒))
3		iotabi 6455	. 2 ⊢ (∀𝑥(𝜓 ↔ 𝜒) → (℩𝑥𝜓) = (℩𝑥𝜒))
4	2, 3	syl 17	1 ⊢ (𝜑 → (℩𝑥𝜓) = (℩𝑥𝜒))

Syntax hints:   → wi 4   ↔ wb 206  ∀wal 1538   = wceq 1540  ℩cio 6440

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3440  df-ss 3922  df-uni 4862  df-iota 6442

This theorem is referenced by:  csbiota  6479  dffv3  6822  fveq1  6825  fveq2  6826  fvres  6845  csbfv12  6872  opabiota  6909  fvco2  6924  fvopab5  6967  riotaeqdv  7311  riotabidv  7312  riotabidva  7329  erov  8748  iunfictbso  10027  isf32lem9  10274  shftval  14999  sumeq1  15614  sumeq2w  15617  sumeq2ii  15618  sumeq2sdv  15628  zsum  15643  isumclim3  15684  isumshft  15764  prodeq1f  15831  prodeq1  15832  prodeq2w  15835  prodeq2ii  15836  prodeq2sdv  15848  zprod  15862  iprodclim3  15925  pcval  16774  grpidval  18553  grpidpropd  18554  gsumvalx  18568  gsumpropd  18570  gsumpropd2lem  18571  gsumress  18574  psgnfval  19397  psgnval  19404  psgndif  21527  dchrptlem1  27191  lgsdchrval  27281  nosupcbv  27630  nosupfv  27634  noinfcbv  27645  noinffv  27649  ajval  30823  adjval  31852  urpropd  33182  resv1r  33287  opprqus0g  33437  prodeq12sdv  36191  cbvsumdavw  36252  cbvproddavw  36253  cbvsumdavw2  36268  cbvproddavw2  36269  bj-finsumval0  37258  uncov  37580  afv2eq12d  47200  funressndmafv2rn  47208  afv2res  47224  dfafv23  47238  afv2co2  47242