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Theorem riotabidva 7407
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3443 analog.) (Contributed by NM, 17-Jan-2012.)
Hypothesis
Ref Expression
riotabidva.1 ((𝜑𝑥𝐴) → (𝜓𝜒))
Assertion
Ref Expression
riotabidva (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Distinct variable group:   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)   𝐴(𝑥)

Proof of Theorem riotabidva
StepHypRef Expression
1 riotabidva.1 . . . 4 ((𝜑𝑥𝐴) → (𝜓𝜒))
21pm5.32da 579 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32iotabidv 6545 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
4 df-riota 7388 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
5 df-riota 7388 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
63, 4, 53eqtr4g 2802 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  cio 6512  crio 7387
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 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-ss 3968  df-uni 4908  df-iota 6514  df-riota 7388
This theorem is referenced by:  riotabiia  7408  dfceil2  13879  cidpropd  17753  grpinvpropd  19033  mirval  28663  mirfv  28664  grpoidval  30532  adjval2  31910  riotaeqbidva  32515  xdivval  32901  toslub  32963  tosglb  32965  ringinvval  33239  glbconN  39378  glbconNOLD  39379  cdlemk33N  40911  cdlemk34  40912  cdlemkid4  40936
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