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Theorem sbcexfi 35574
 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 35572 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3 sbcexfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43exbii 1849 . 2 (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
52, 4bitri 278 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∃wex 1781  Ⅎwnfc 2936  [wsbc 3720 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-v 3443  df-sbc 3721 This theorem is referenced by: (None)
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