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Theorem sbcexfi 38124
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 38122 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3 sbcexfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43exbii 1848 . 2 (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
52, 4bitri 275 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1779  wnfc 2890  [wsbc 3788
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-v 3482  df-sbc 3789
This theorem is referenced by: (None)
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