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Theorem alxfr 4159
Description: Transfer universal quantification from a variable x to another variable y contained in expression A. (Contributed by NM, 18-Feb-2007.)
Hypothesis
Ref Expression
alxfr.1 (x = A → (φψ))
Assertion
Ref Expression
alxfr ((y A B xy x = A) → (xφyψ))
Distinct variable groups:   x,A   φ,y   ψ,x   x,y
Allowed substitution hints:   φ(x)   ψ(y)   A(y)   B(x,y)

Proof of Theorem alxfr
StepHypRef Expression
1 alxfr.1 . . . . . . 7 (x = A → (φψ))
21spcgv 2634 . . . . . 6 (A B → (xφψ))
32com12 27 . . . . 5 (xφ → (A Bψ))
43alimdv 1756 . . . 4 (xφ → (y A Byψ))
54com12 27 . . 3 (y A B → (xφyψ))
65adantr 261 . 2 ((y A B xy x = A) → (xφyψ))
7 nfa1 1431 . . . . . 6 yyψ
8 nfv 1418 . . . . . 6 yφ
9 sp 1398 . . . . . . 7 (yψψ)
109, 1syl5ibrcom 146 . . . . . 6 (yψ → (x = Aφ))
117, 8, 10exlimd 1485 . . . . 5 (yψ → (y x = Aφ))
1211alimdv 1756 . . . 4 (yψ → (xy x = Axφ))
1312com12 27 . . 3 (xy x = A → (yψxφ))
1413adantl 262 . 2 ((y A B xy x = A) → (yψxφ))
156, 14impbid 120 1 ((y A B xy x = A) → (xφyψ))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390
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-bnd 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-v 2553
This theorem is referenced by: (None)
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