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Theorem gencbvex2 2595
 Description: Restatement of gencbvex 2594 with weaker hypotheses. (Contributed by Jeff Hankins, 6-Dec-2006.)
Hypotheses
Ref Expression
gencbvex2.1 A V
gencbvex2.2 (A = y → (φψ))
gencbvex2.3 (A = y → (χθ))
gencbvex2.4 (θx(χ A = y))
Assertion
Ref Expression
gencbvex2 (x(χ φ) ↔ y(θ ψ))
Distinct variable groups:   ψ,x   φ,y   θ,x   χ,y   y,A
Allowed substitution hints:   φ(x)   ψ(y)   χ(x)   θ(y)   A(x)

Proof of Theorem gencbvex2
StepHypRef Expression
1 gencbvex2.1 . 2 A V
2 gencbvex2.2 . 2 (A = y → (φψ))
3 gencbvex2.3 . 2 (A = y → (χθ))
4 gencbvex2.4 . . 3 (θx(χ A = y))
53biimpac 282 . . . 4 ((χ A = y) → θ)
65exlimiv 1486 . . 3 (x(χ A = y) → θ)
74, 6impbii 117 . 2 (θx(χ A = y))
81, 2, 3, 7gencbvex 2594 1 (x(χ φ) ↔ y(θ ψ))
 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: (None)
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