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Theorem cbvraldva2 2531
Description: Rule used to change the bound variable in a restricted universal 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
cbvraldva2 (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 cbvraldva2
StepHypRef Expression
1 simpr 103 . . . . 5 ((φ x = y) → x = y)
2 cbvraldva2.2 . . . . 5 ((φ x = y) → A = B)
31, 2eleq12d 2105 . . . 4 ((φ x = y) → (x Ay B))
4 cbvraldva2.1 . . . 4 ((φ x = y) → (ψχ))
53, 4imbi12d 223 . . 3 ((φ x = y) → ((x Aψ) ↔ (y Bχ)))
65cbvaldva 1800 . 2 (φ → (x(x Aψ) ↔ y(y Bχ)))
7 df-ral 2305 . 2 (x A ψx(x Aψ))
8 df-ral 2305 . 2 (y B χy(y Bχ))
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  wal 1240   = wceq 1242   wcel 1390  wral 2300
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-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-cleq 2030  df-clel 2033  df-ral 2305
This theorem is referenced by:  cbvraldva  2533  acexmid  5454  tfrlem3ag  5865  tfrlem3a  5866  tfrlemi1  5887
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