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Theorem spesbc 3660
Description: Existence form of spsbc 3587. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 3584 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 rspesbca 3659 . . 3 ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
31, 2mpancom 706 . 2 ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4 rexv 3358 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
53, 4sylib 208 1 ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1851  wcel 2137  wrex 3049  Vcvv 3338  [wsbc 3574
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ral 3053  df-rex 3054  df-v 3340  df-sbc 3575
This theorem is referenced by:  spesbcd  3661  opelopabsb  5133  sbccomieg  37857  frege124d  38553  sbiota1  39135
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