ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ceqsex GIF version

Theorem ceqsex 2657
Description: Elimination of an existential quantifier, using implicit substitution. (Contributed by NM, 2-Mar-1995.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypotheses
Ref Expression
ceqsex.1 𝑥𝜓
ceqsex.2 𝐴 ∈ V
ceqsex.3 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
ceqsex (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem ceqsex
StepHypRef Expression
1 ceqsex.1 . . 3 𝑥𝜓
2 ceqsex.3 . . . 4 (𝑥 = 𝐴 → (𝜑𝜓))
32biimpa 290 . . 3 ((𝑥 = 𝐴𝜑) → 𝜓)
41, 3exlimi 1530 . 2 (∃𝑥(𝑥 = 𝐴𝜑) → 𝜓)
52biimprcd 158 . . . 4 (𝜓 → (𝑥 = 𝐴𝜑))
61, 5alrimi 1460 . . 3 (𝜓 → ∀𝑥(𝑥 = 𝐴𝜑))
7 ceqsex.2 . . . 4 𝐴 ∈ V
87isseti 2627 . . 3 𝑥 𝑥 = 𝐴
9 exintr 1570 . . 3 (∀𝑥(𝑥 = 𝐴𝜑) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝑥 = 𝐴𝜑)))
106, 8, 9mpisyl 1380 . 2 (𝜓 → ∃𝑥(𝑥 = 𝐴𝜑))
114, 10impbii 124 1 (∃𝑥(𝑥 = 𝐴𝜑) ↔ 𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1287   = wceq 1289  wnf 1394  wex 1426  wcel 1438  Vcvv 2619
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-v 2621
This theorem is referenced by:  ceqsexv  2658  ceqsex2  2659
  Copyright terms: Public domain W3C validator