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Theorem cbvrexdva2 2512
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 103 . . . . 5 ((φ x = y) → x = y)
2 cbvraldva2.2 . . . . 5 ((φ x = y) → A = B)
31, 2eleq12d 2086 . . . 4 ((φ x = y) → (x Ay B))
4 cbvraldva2.1 . . . 4 ((φ x = y) → (ψχ))
53, 4anbi12d 445 . . 3 ((φ x = y) → ((x A ψ) ↔ (y B χ)))
65cbvexdva 1782 . 2 (φ → (x(x A ψ) ↔ y(y B χ)))
7 df-rex 2286 . 2 (x A ψx(x A ψ))
8 df-rex 2286 . 2 (y B χy(y B χ))
96, 7, 83bitr4g 212 1 (φ → (x A ψy B χ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1226  wex 1358   wcel 1370  wrex 2281
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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-nf 1326  df-cleq 2011  df-clel 2014  df-rex 2286
This theorem is referenced by:  cbvrexdva  2514  acexmid  5431
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