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Theorem sbcexfi 32890
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 32888 . 2 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
3 sbcexfi.2 . . 3 ([𝐴 / 𝑥]𝜑𝜓)
43exbii 1763 . 2 (∃𝑦[𝐴 / 𝑥]𝜑 ↔ ∃𝑦𝜓)
52, 4bitri 262 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦𝜓)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wex 1694  wnfc 2734  [wsbc 3398
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2229  ax-ext 2586
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2593  df-cleq 2599  df-clel 2602  df-nfc 2736  df-v 3171  df-sbc 3399
This theorem is referenced by: (None)
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