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Theorem cbvrexdva2 2840
 Description: Rule used to change the bound variable in a restricted existential quantifier with implicit substitution which also changes the quantifier domain. Deduction form. (Contributed by David Moews, 1-May-2017.)
Hypotheses
Ref Expression
cbvraldva2.1 ((φ x = y) → (ψχ))
cbvraldva2.2 ((φ x = y) → A = B)
Assertion
Ref Expression
cbvrexdva2 (φ → (x A ψy B χ))
Distinct variable groups:   y,A   ψ,y   x,B   χ,x   φ,x,y
Allowed substitution hints:   ψ(x)   χ(y)   A(x)   B(y)

Proof of Theorem cbvrexdva2
StepHypRef Expression
1 simpr 447 . . . . 5 ((φ x = y) → x = y)
2 cbvraldva2.2 . . . . 5 ((φ x = y) → A = B)
31, 2eleq12d 2421 . . . 4 ((φ x = y) → (x Ay B))
4 cbvraldva2.1 . . . 4 ((φ x = y) → (ψχ))
53, 4anbi12d 691 . . 3 ((φ x = y) → ((x A ψ) ↔ (y B χ)))
65cbvexdva 2011 . 2 (φ → (x(x A ψ) ↔ y(y B χ)))
7 df-rex 2620 . 2 (x A ψx(x A ψ))
8 df-rex 2620 . 2 (y B χy(y B χ))
96, 7, 83bitr4g 279 1 (φ → (x A ψy B χ))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  ∃wrex 2615 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-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-cleq 2346  df-clel 2349  df-rex 2620 This theorem is referenced by:  cbvrexdva  2842
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