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Theorem sbcexfi 33090
 Description: Move existential quantifier in and out of class substitution, with an explicit non-free variable condition and in inference form. (Contributed by Giovanni Mascellani, 30-May-2019.)
Hypotheses
Ref Expression
sbcexfi.1 𝑦𝐴
sbcexfi.2 ([𝐴 / 𝑥]𝜑𝜓)
Assertion
Ref Expression
sbcexfi ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbcexfi
StepHypRef Expression
1 sbcexfi.1 . . 3 𝑦𝐴
21sbcexf 33088 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3 sbcexfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43exbii 1764 . 2 (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
52, 4bitri 263 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195  ∃wex 1695  Ⅎwnfc 2738  [wsbc 3402 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-sbc 3403 This theorem is referenced by: (None)
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