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Theorem nfralw 2524
Description: Bound-variable hypothesis builder for restricted quantification. See nfralya 2527 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 1476 . . 3 𝑦
2 nfralw.1 . . . 4 𝑥𝐴
32a1i 9 . . 3 (⊤ → 𝑥𝐴)
4 nfralw.2 . . . 4 𝑥𝜑
54a1i 9 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfraldw 2519 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1372 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff set class
Syntax hints:  wtru 1364  wnf 1470  wnfc 2316  wral 2465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-4 1520  ax-17 1536  ax-ial 1544  ax-i5r 1545  ax-ext 2169
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-nf 1471  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ral 2470
This theorem is referenced by:  rspc2vd  3137  fprod2dlemstep  11643  fprodcom2fi  11647  nnwofdc  12052
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