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Theorem riotabidva 5890
Description: Equivalent wff's yield equal restricted class abstractions (deduction form). (rabbidva 2748 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 452 . . 3 (𝜑 → ((𝑥𝐴𝜓) ↔ (𝑥𝐴𝜒)))
32iotabidv 5237 . 2 (𝜑 → (℩𝑥(𝑥𝐴𝜓)) = (℩𝑥(𝑥𝐴𝜒)))
4 df-riota 5873 . 2 (𝑥𝐴 𝜓) = (℩𝑥(𝑥𝐴𝜓))
5 df-riota 5873 . 2 (𝑥𝐴 𝜒) = (℩𝑥(𝑥𝐴𝜒))
63, 4, 53eqtr4g 2251 1 (𝜑 → (𝑥𝐴 𝜓) = (𝑥𝐴 𝜒))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105   = wceq 1364  wcel 2164  cio 5213  crio 5872
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-uni 3836  df-iota 5215  df-riota 5873
This theorem is referenced by:  riotabiia  5891  divfnzn  9686  grpinvpropdg  13147  oppr1g  13578
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