Theorem issetf 2865

Description: A version of isset 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	⊢ ℲxA

Assertion

Ref	Expression
issetf	⊢ (A ∈ V ↔ ∃x x = A)

Proof of Theorem issetf

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isset 2864	. 2 ⊢ (A ∈ V ↔ ∃y y = A)
2		issetf.1	. . . 4 ⊢ ℲxA
3	2	nfeq2 2501	. . 3 ⊢ Ⅎx y = A
4		nfv 1619	. . 3 ⊢ Ⅎy x = A
5		eqeq1 2359	. . 3 ⊢ (y = x → (y = A ↔ x = A))
6	3, 4, 5	cbvex 1985	. 2 ⊢ (∃y y = A ↔ ∃x x = A)
7	1, 6	bitri 240	1 ⊢ (A ∈ V ↔ ∃x x = A)

Syntax hints:   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Ⅎwnfc 2477  Vcvv 2860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is referenced by:  vtoclgf  2914  spcimgft  2931