Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rspcegf Structured version   Visualization version   GIF version

Theorem rspcegf 45028
Description: A version of rspcev 3622 using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 20-Apr-2017.)
Hypotheses
Ref Expression
rspcegf.1 𝑥𝜓
rspcegf.2 𝑥𝐴
rspcegf.3 𝑥𝐵
rspcegf.4 (𝑥 = 𝐴 → (𝜑𝜓))
Assertion
Ref Expression
rspcegf ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)

Proof of Theorem rspcegf
StepHypRef Expression
1 rspcegf.2 . . . 4 𝑥𝐴
2 rspcegf.3 . . . . . 6 𝑥𝐵
31, 2nfel 2920 . . . . 5 𝑥 𝐴𝐵
4 rspcegf.1 . . . . 5 𝑥𝜓
53, 4nfan 1899 . . . 4 𝑥(𝐴𝐵𝜓)
6 eleq1 2829 . . . . 5 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
7 rspcegf.4 . . . . 5 (𝑥 = 𝐴 → (𝜑𝜓))
86, 7anbi12d 632 . . . 4 (𝑥 = 𝐴 → ((𝑥𝐵𝜑) ↔ (𝐴𝐵𝜓)))
91, 5, 8spcegf 3592 . . 3 (𝐴𝐵 → ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑)))
109anabsi5 669 . 2 ((𝐴𝐵𝜓) → ∃𝑥(𝑥𝐵𝜑))
11 df-rex 3071 . 2 (∃𝑥𝐵 𝜑 ↔ ∃𝑥(𝑥𝐵𝜑))
1210, 11sylibr 234 1 ((𝐴𝐵𝜓) → ∃𝑥𝐵 𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wex 1779  wnf 1783  wcel 2108  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-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-rex 3071
This theorem is referenced by:  rspcef  45077  stoweidlem46  46061
  Copyright terms: Public domain W3C validator