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Theorem nfrex 3375
Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2377. See nfrexw 3313 for a version with a disjoint variable condition, but not requiring ax-13 2377. (Contributed by NM, 1-Sep-1999.) (Revised by Mario Carneiro, 7-Oct-2016.) (Proof shortened by Wolf Lammen, 30-Dec-2019.) (New usage is discouraged.)
Hypotheses
Ref Expression
nfral.1 𝑥𝐴
nfral.2 𝑥𝜑
Assertion
Ref Expression
nfrex 𝑥𝑦𝐴 𝜑

Proof of Theorem nfrex
StepHypRef Expression
1 nftru 1804 . . 3 𝑦
2 nfral.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfral.2 . . . 4 𝑥𝜑
54a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexd 3373 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1547 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1541  wnf 1783  wnfc 2890  wrex 3070
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-13 2377  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-nf 1784  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071
This theorem is referenced by:  nfiung  5025
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