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Theorem spc2ev 2947
 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 2941 . 2 ((A V B V) → (ψxyφ))
51, 2, 4mp2an 653 1 (ψxyφ)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2859 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-v 2861 This theorem is referenced by:  opkabssvvk  4208  dfpw12  4301  pw1equn  4331  pw1eqadj  4332  dfxp2  5113  brtxp  5783  endisj  6051  mucnc  6131  ce0nnul  6177  cenc  6181  ce2  6192
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