MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  spesbc Structured version   Visualization version   GIF version

Theorem spesbc 3727
Description: Existence form of spsbc 3657. (Contributed by Mario Carneiro, 18-Nov-2016.)
Assertion
Ref Expression
spesbc ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)

Proof of Theorem spesbc
StepHypRef Expression
1 sbcex 3654 . . 3 ([𝐴 / 𝑥]𝜑𝐴 ∈ V)
2 rspesbca 3726 . . 3 ((𝐴 ∈ V ∧ [𝐴 / 𝑥]𝜑) → ∃𝑥 ∈ V 𝜑)
31, 2mpancom 671 . 2 ([𝐴 / 𝑥]𝜑 → ∃𝑥 ∈ V 𝜑)
4 rexv 3425 . 2 (∃𝑥 ∈ V 𝜑 ↔ ∃𝑥𝜑)
53, 4sylib 209 1 ([𝐴 / 𝑥]𝜑 → ∃𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1859  wcel 2157  wrex 3108  Vcvv 3402  [wsbc 3644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ral 3112  df-rex 3113  df-v 3404  df-sbc 3645
This theorem is referenced by:  spesbcd  3728  opelopabsb  5191  sbccomieg  37864  frege124d  38558  sbiota1  39139
  Copyright terms: Public domain W3C validator