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Theorem spc2ev 2706
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 2700 . 2 ((𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝜓 → ∃𝑥𝑦𝜑))
51, 2, 4mp2an 417 1 (𝜓 → ∃𝑥𝑦𝜑)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1287  wex 1424  wcel 1436  Vcvv 2614
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-v 2616
This theorem is referenced by:  relop  4547  th3qlem2  6328  endisj  6473  axcnre  7337
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