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Theorem sbcexfi 36012
Description: Move existential quantifier in and out of class substitution, with an explicit nonfree 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 36010 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3 sbcexfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43exbii 1855 . 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 1787  wnfc 2884  [wsbc 3694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410  df-sbc 3695
This theorem is referenced by: (None)
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