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Theorem spc2ev 2642
 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 = A y = B) → (φψ))
Assertion
Ref Expression
spc2ev (ψxyφ)
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 = A y = B) → (φψ))
43spc2egv 2636 . 2 ((A V B V) → (ψxyφ))
51, 2, 4mp2an 402 1 (ψxyφ)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-v 2553 This theorem is referenced by:  relop  4429  th3qlem2  6145  endisj  6234  axcnre  6765
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