Theorem rspce 2951

Description: Restricted existential specialization, using implicit substitution. (Contributed by NM, 26-May-1998.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

Ref	Expression
rspc.1	⊢ Ⅎxψ
rspc.2	⊢ (x = A → (φ ↔ ψ))

Assertion

Ref	Expression
rspce	⊢ ((A ∈ B ∧ ψ) → ∃x ∈ B φ)

Distinct variable groups:   x,A   x,B

Allowed substitution hints:   φ(x)   ψ(x)

Proof of Theorem rspce

Step	Hyp	Ref	Expression
1		nfcv 2490	. . . 4 ⊢ ℲxA
2		nfv 1619	. . . . 5 ⊢ Ⅎx A ∈ B
3		rspc.1	. . . . 5 ⊢ Ⅎxψ
4	2, 3	nfan 1824	. . . 4 ⊢ Ⅎx(A ∈ B ∧ ψ)
5		eleq1 2413	. . . . 5 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
6		rspc.2	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
7	5, 6	anbi12d 691	. . . 4 ⊢ (x = A → ((x ∈ B ∧ φ) ↔ (A ∈ B ∧ ψ)))
8	1, 4, 7	spcegf 2936	. . 3 ⊢ (A ∈ B → ((A ∈ B ∧ ψ) → ∃x(x ∈ B ∧ φ)))
9	8	anabsi5 790	. 2 ⊢ ((A ∈ B ∧ ψ) → ∃x(x ∈ B ∧ φ))
10		df-rex 2621	. 2 ⊢ (∃x ∈ B φ ↔ ∃x(x ∈ B ∧ φ))
11	9, 10	sylibr 203	1 ⊢ ((A ∈ B ∧ ψ) → ∃x ∈ B φ)

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∃wrex 2616

This proof depends on 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 proof 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-rex 2621  df-v 2862

This theorem is used by:  rspcev  2956