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Theorem spesbc 3872
Description: Existence form of spsbc 3787. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 3784 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 rspesbca 3871 . . 3 ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
31, 2mpancom 687 . 2 ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4 rexv 3495 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
53, 4sylib 217 1 ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1774  wcel 2099  wrex 3065  Vcvv 3469  [wsbc 3774
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3057  df-rex 3066  df-v 3471  df-sbc 3775
This theorem is referenced by:  spesbcd  3873  opelopabsb  5526  sbccomieg  42135  frege124d  43114  sbiota1  43794
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