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

Theorem issetf 3180
Description: A version of isset 3179 that does not require 𝑥 and 𝐴 to be distinct. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 10-Oct-2016.)
Hypothesis
Ref Expression
issetf.1 𝑥𝐴
Assertion
Ref Expression
issetf (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)

Proof of Theorem issetf
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 isset 3179 . 2 (𝐴 ∈ V ↔ ∃𝑦 𝑦 = 𝐴)
2 issetf.1 . . . 4 𝑥𝐴
32nfeq2 2765 . . 3 𝑥 𝑦 = 𝐴
4 nfv 1829 . . 3 𝑦 𝑥 = 𝐴
5 eqeq1 2613 . . 3 (𝑦 = 𝑥 → (𝑦 = 𝐴𝑥 = 𝐴))
63, 4, 5cbvex 2259 . 2 (∃𝑦 𝑦 = 𝐴 ↔ ∃𝑥 𝑥 = 𝐴)
71, 6bitri 262 1 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 194   = wceq 1474  wex 1694  wcel 1976  wnfc 2737  Vcvv 3172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174
This theorem is referenced by:  vtoclgf  3236  spcimgft  3256
  Copyright terms: Public domain W3C validator