MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ceqsexv2d Structured version   Visualization version   GIF version

Theorem ceqsexv2d 3488
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by Thierry Arnoux, 10-Sep-2016.)
Hypotheses
Ref Expression
ceqsexv2d.1 𝐴 ∈ V
ceqsexv2d.2 (𝑥 = 𝐴 → (𝜑𝜓))
ceqsexv2d.3 𝜓
Assertion
Ref Expression
ceqsexv2d 𝑥𝜑
Distinct variable groups:   𝑥,𝐴   𝜓,𝑥
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem ceqsexv2d
StepHypRef Expression
1 ceqsexv2d.3 . 2 𝜓
2 ceqsexv2d.1 . . . 4 𝐴 ∈ V
3 ceqsexv2d.2 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
42, 3ceqsexv 3487 . . 3 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
54biimpri 229 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
6 exsimpr 1855 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → ∃𝑥𝜑)
71, 5, 6mp2b 10 1 𝑥𝜑
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1525  wex 1765  wcel 2083  Vcvv 3440
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-12 2143  ax-ext 2771
This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1766  df-nf 1770  df-cleq 2790  df-clel 2865
This theorem is referenced by:  ensn1  8428  tz9.1  9024  cplem2  9172  karden  9177  2lgslem1  25656  griedg0prc  26733  1loopgrvd2  26972  bnj150  31760  fnchoice  40846  nfermltl8rev  43411  nfermltl2rev  43412  nfermltlrev  43413  rrx2xpreen  44209
  Copyright terms: Public domain W3C validator