Theorem rspc 2950

Description: Restricted specialization, using implicit substitution. (Contributed by NM, 19-Apr-2005.) (Revised by Mario Carneiro, 11-Oct-2016.)

Hypotheses

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

Assertion

Ref	Expression
rspc	⊢ (A ∈ B → (∀x ∈ B φ → ψ))

Distinct variable groups:   x,A   x,B

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

Proof of Theorem rspc

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀x ∈ B φ ↔ ∀x(x ∈ B → φ))
2		nfcv 2490	. . . 4 ⊢ ℲxA
3		nfv 1619	. . . . 5 ⊢ Ⅎx A ∈ B
4		rspc.1	. . . . 5 ⊢ Ⅎxψ
5	3, 4	nfim 1813	. . . 4 ⊢ Ⅎx(A ∈ B → ψ)
6		eleq1 2413	. . . . 5 ⊢ (x = A → (x ∈ B ↔ A ∈ B))
7		rspc.2	. . . . 5 ⊢ (x = A → (φ ↔ ψ))
8	6, 7	imbi12d 311	. . . 4 ⊢ (x = A → ((x ∈ B → φ) ↔ (A ∈ B → ψ)))
9	2, 5, 8	spcgf 2935	. . 3 ⊢ (A ∈ B → (∀x(x ∈ B → φ) → (A ∈ B → ψ)))
10	9	pm2.43a 45	. 2 ⊢ (A ∈ B → (∀x(x ∈ B → φ) → ψ))
11	1, 10	syl5bi 208	1 ⊢ (A ∈ B → (∀x ∈ B φ → ψ))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540  Ⅎwnf 1544   = wceq 1642   ∈ wcel 1710  ∀wral 2615

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-ral 2620  df-v 2862

This theorem is referenced by:  rspcv  2952  rspc2  2961