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Theorem ralxfr 4164
Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 10-Jun-2005.) (Revised by Mario Carneiro, 15-Aug-2014.)
Hypotheses
Ref Expression
ralxfr.1 (y 𝐶A B)
ralxfr.2 (x By 𝐶 x = A)
ralxfr.3 (x = A → (φψ))
Assertion
Ref Expression
ralxfr (x B φy 𝐶 ψ)
Distinct variable groups:   ψ,x   φ,y   x,A   x,y,B   x,𝐶
Allowed substitution hints:   φ(x)   ψ(y)   A(y)   𝐶(y)

Proof of Theorem ralxfr
StepHypRef Expression
1 ralxfr.1 . . . 4 (y 𝐶A B)
21adantl 262 . . 3 (( ⊤ y 𝐶) → A B)
3 ralxfr.2 . . . 4 (x By 𝐶 x = A)
43adantl 262 . . 3 (( ⊤ x B) → y 𝐶 x = A)
5 ralxfr.3 . . . 4 (x = A → (φψ))
65adantl 262 . . 3 (( ⊤ x = A) → (φψ))
72, 4, 6ralxfrd 4160 . 2 ( ⊤ → (x B φy 𝐶 ψ))
87trud 1251 1 (x B φy 𝐶 ψ)
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  wtru 1243   wcel 1390  wral 2300  wrex 2301
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-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553
This theorem is referenced by: (None)
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