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

Theorem spesbcd 3508
Description: form of spsbc 3435. (Contributed by Mario Carneiro, 9-Feb-2017.)
Hypothesis
Ref Expression
spesbcd.1 (𝜑[𝐴 / 𝑥]𝜓)
Assertion
Ref Expression
spesbcd (𝜑 → ∃𝑥𝜓)

Proof of Theorem spesbcd
StepHypRef Expression
1 spesbcd.1 . 2 (𝜑[𝐴 / 𝑥]𝜓)
2 spesbc 3507 . 2 ([𝐴 / 𝑥]𝜓 → ∃𝑥𝜓)
31, 2syl 17 1 (𝜑 → ∃𝑥𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wex 1701  [wsbc 3422
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-sbc 3423
This theorem is referenced by:  euotd  4945  ex-natded9.26  27164  bnj1465  30676  bj-sels  32650  spesbcdi  33596  brtrclfv2  37539  cotrclrcl  37554
  Copyright terms: Public domain W3C validator