Theorem spcev 2947

Description: Existential specialization, using implicit substitution. (Contributed by NM, 31-Dec-1993.) (Proof shortened by Eric Schmidt, 22-Dec-2006.)

Hypotheses

Ref	Expression
spcv.1	⊢ A ∈ V
spcv.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
spcev	⊢ (ψ → ∃xφ)

Distinct variable groups:   x,A   ψ,x

Allowed substitution hint:   φ(x)

Proof of Theorem spcev

Step	Hyp	Ref	Expression
1		spcv.1	. 2 ⊢ A ∈ V
2		spcv.2	. . 3 ⊢ (x = A → (φ ↔ ψ))
3	2	spcegv 2941	. 2 ⊢ (A ∈ V → (ψ → ∃xφ))
4	1, 3	ax-mp 5	1 ⊢ (ψ → ∃xφ)

Syntax hints:   → wi 4   ↔ wb 176  ∃wex 1541   = wceq 1642   ∈ wcel 1710  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:  snel1c  4141  sspw1  4336  sspw12  4337  sfin01  4529  sfindbl  4531  1cvsfin  4543  vfinspsslem1  4551  phialllem1  4617  phialllem2  4618  nfunv  5139  ffoss  5315  map0  6026  unen  6049  enpw1  6063  1cnc  6140  ncaddccl  6145  ce0addcnnul  6180  ce0nulnc  6185  dflec3  6222  nclenc  6223