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Theorem riotabidva 6853
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 3370 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 575 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32iotabidv 6083 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
4 df-riota 6837 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
5 df-riota 6837 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
63, 4, 53eqtr4g 2856 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  cio 6060  crio 6836
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-rex 3093  df-uni 4627  df-iota 6062  df-riota 6837
This theorem is referenced by:  riotabiia  6854  dfceil2  12891  cidpropd  16681  grpinvpropd  17803  mirval  25899  mirfv  25900  grpoidval  27885  adjval2  29267  xdivval  30135  toslub  30176  tosglb  30178  ringinvval  30300  glbconN  35390  cdlemk33N  36922  cdlemk34  36923  cdlemkid4  36947
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