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Theorem spc2ev 2665
Description: Existential specialization, using implicit substitution. (Contributed by NM, 3-Aug-1995.)
Hypotheses
Ref Expression
spc2ev.1 𝐴 ∈ V
spc2ev.2 𝐵 ∈ V
spc2ev.3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
Assertion
Ref Expression
spc2ev (𝜓 → ∃𝑥𝑦𝜑)
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝜓,𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)

Proof of Theorem spc2ev
StepHypRef Expression
1 spc2ev.1 . 2 𝐴 ∈ V
2 spc2ev.2 . 2 𝐵 ∈ V
3 spc2ev.3 . . 3 ((𝑥 = 𝐴𝑦 = 𝐵) → (𝜑𝜓))
43spc2egv 2659 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥𝑦𝜑))
51, 2, 4mp2an 410 1 (𝜓 → ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wex 1397  wcel 1409  Vcvv 2574
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-v 2576
This theorem is referenced by:  relop  4513  th3qlem2  6239  endisj  6328  axcnre  7012
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