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Theorem spesbc 3738
Description: Existence form of spsbc 3665. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 3662 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 rspesbca 3737 . . 3 ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
31, 2mpancom 678 . 2 ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4 rexv 3422 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
53, 4sylib 210 1 ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1823  wcel 2107  wrex 3091  Vcvv 3398  [wsbc 3652
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-v 3400  df-sbc 3653
This theorem is referenced by:  spesbcd  3739  opelopabsb  5222  sbccomieg  38317  frege124d  39010  sbiota1  39590
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