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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 𝑥𝐴
nfralw.2 𝑥𝜑
Assertion
Ref Expression
nfralw 𝑥𝑦𝐴 𝜑
Distinct variable group:   𝑥,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfralw
StepHypRef Expression
1 nftru 1459 . . 3 𝑦
2 nfralw.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralw.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldw 2502 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1357 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1349  wnf 1453  wnfc 2299  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  11585  fprodcom2fi  11589  nnwofdc  11993
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