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Theorem sbcexf 34340
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 2009 . . . 4 𝑧𝜑
21sb8e 2516 . . 3 (∃𝑦𝜑 ↔ ∃𝑧[𝑧 / 𝑦]𝜑)
32sbcbii 3652 . 2 ([𝐴 / 𝑥]𝑦𝜑[𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑)
4 sbcex2 3647 . 2 ([𝐴 / 𝑥]𝑧[𝑧 / 𝑦]𝜑 ↔ ∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑)
5 sbcexf.1 . . . 4 𝑦𝐴
6 nfs1v 2287 . . . 4 𝑦[𝑧 / 𝑦]𝜑
75, 6nfsbc 3618 . . 3 𝑦[𝐴 / 𝑥][𝑧 / 𝑦]𝜑
8 nfv 2009 . . 3 𝑧[𝐴 / 𝑥]𝜑
9 sbequ12r 2279 . . . 4 (𝑧 = 𝑦 → ([𝑧 / 𝑦]𝜑𝜑))
109sbcbidv 3651 . . 3 (𝑧 = 𝑦 → ([𝐴 / 𝑥][𝑧 / 𝑦]𝜑[𝐴 / 𝑥]𝜑))
117, 8, 10cbvex 2377 . 2 (∃𝑧[𝐴 / 𝑥][𝑧 / 𝑦]𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
123, 4, 113bitri 288 1 ([𝐴 / 𝑥]𝑦𝜑 ↔ ∃𝑦[𝐴 / 𝑥]𝜑)
Colors of variables: wff setvar class
Syntax hints:  wb 197  wex 1874  [wsb 2062  wnfc 2894  [wsbc 3596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-sbc 3597
This theorem is referenced by:  sbcexfi  34342
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