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Theorem riotabidva 7387
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3437 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 577 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32iotabidv 6526 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
4 df-riota 7367 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
5 df-riota 7367 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
63, 4, 53eqtr4g 2795 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394   = wceq 1539  wcel 2104  cio 6492  crio 7366
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-uni 4908  df-iota 6494  df-riota 7367
This theorem is referenced by:  riotabiia  7388  dfceil2  13808  cidpropd  17658  grpinvpropd  18934  mirval  28173  mirfv  28174  grpoidval  30033  adjval2  31411  riotaeqbidva  32003  xdivval  32352  toslub  32410  tosglb  32412  ringinvval  32654  glbconN  38550  glbconNOLD  38551  cdlemk33N  40083  cdlemk34  40084  cdlemkid4  40108
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