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Theorem spesbc 3813
 Description: Existence form of spsbc 3735. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 3732 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 rspesbca 3812 . . 3 ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
31, 2mpancom 687 . 2 ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4 rexv 3468 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
53, 4sylib 221 1 ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∃wex 1781   ∈ wcel 2111  ∃wrex 3107  Vcvv 3442  [wsbc 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3444  df-sbc 3723 This theorem is referenced by:  spesbcd  3814  opelopabsb  5386  sbccomieg  39905  frege124d  40633  sbiota1  41309
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