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Theorem nfralw 2507
Description: Bound-variable hypothesis builder for restricted quantification. See nfralya 2510 for a version with  y and 
A distinct instead of  x and  y. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.)
Hypotheses
Ref Expression
nfralw.1  |-  F/_ x A
nfralw.2  |-  F/ x ph
Assertion
Ref Expression
nfralw  |-  F/ x A. y  e.  A  ph
Distinct variable group:    x, y
Allowed substitution hints:    ph( x, y)    A( x, y)

Proof of Theorem nfralw
StepHypRef Expression
1 nftru 1459 . . 3  |-  F/ y T.
2 nfralw.1 . . . 4  |-  F/_ x A
32a1i 9 . . 3  |-  ( T. 
->  F/_ x A )
4 nfralw.2 . . . 4  |-  F/ x ph
54a1i 9 . . 3  |-  ( T. 
->  F/ x ph )
61, 3, 5nfraldw 2502 . 2  |-  ( T. 
->  F/ x A. y  e.  A  ph )
76mptru 1357 1  |-  F/ x A. y  e.  A  ph
Colors of variables: wff set class
Syntax hints:   T. wtru 1349   F/wnf 1453   F/_wnfc 2299   A.wral 2448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-4 1503  ax-17 1519  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453
This theorem is referenced by:  rspc2vd  3117  fprod2dlemstep  11578  fprodcom2fi  11582  nnwofdc  11986
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