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Theorem sbcexf 32882
Description: Move existential quantifier in and out of class substitution, with an explicit non-free variable condition. (Contributed by Giovanni Mascellani, 29-May-2019.)
Hypothesis
Ref Expression
sbcexf.1 𝑦𝐴
Assertion
Ref Expression
sbcexf ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem sbcexf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 nfv 1830 . . . 4 𝑧𝜑
21sb8e 2413 . . 3 (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑)
32sbcbii 3458 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
4 sbcex2 3453 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 sbcexf.1 . . . 4 𝑦𝐴
6 nfs1v 2425 . . . 4 𝑦[𝑧 / 𝑦]𝜑
75, 6nfsbc 3424 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8 nfv 1830 . . 3 𝑧[𝐴 / 𝑥]𝜑
9 sbequ12r 2098 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
109sbcbidv 3457 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
117, 8, 10cbvex 2260 . 2 (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
123, 4, 113bitri 285 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 195   = wceq 1475  wex 1695  [wsb 1867  wnfc 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:  sbcexfi  32884
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