MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nfrex Structured version   Visualization version   GIF version

Theorem nfrex 3373
Description: Bound-variable hypothesis builder for restricted quantification. Usage of this theorem is discouraged because it depends on ax-13 2375. See nfrexw 3311 for a version with a disjoint variable condition, but not requiring ax-13 2375. (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 1801 . . 3 𝑦
2 nfral.1 . . . 4 𝑥𝐴
32a1i 11 . . 3 (⊤ → 𝑥𝐴)
4 nfral.2 . . . 4 𝑥𝜑
54a1i 11 . . 3 (⊤ → Ⅎ𝑥𝜑)
61, 3, 5nfrexd 3371 . 2 (⊤ → Ⅎ𝑥𝑦𝐴 𝜑)
76mptru 1544 1 𝑥𝑦𝐴 𝜑
Colors of variables: wff setvar class
Syntax hints:  wtru 1538  wnf 1780  wnfc 2888  wrex 3068
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-13 2375  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-nf 1781  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069
This theorem is referenced by:  nfiung  5030
  Copyright terms: Public domain W3C validator