Theorem rspcimdv 2957

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.)

Hypotheses

Ref	Expression
rspcimdv.1	⊢ (φ → A ∈ B)
rspcimdv.2	⊢ ((φ ∧ x = A) → (ψ → χ))

Assertion

Ref	Expression
rspcimdv	⊢ (φ → (∀x ∈ B ψ → χ))

Distinct variable groups:   x,A   x,B   φ,x   χ,x

Allowed substitution hint:   ψ(x)

Proof of Theorem rspcimdv

Step	Hyp	Ref	Expression
1		df-ral 2620	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
2		rspcimdv.1	. . 3 ⊢ (φ → A ∈ B)
3		simpr 447	. . . . . . 7 ⊢ ((φ ∧ x = A) → x = A)
4	3	eleq1d 2419	. . . . . 6 ⊢ ((φ ∧ x = A) → (x ∈ B ↔ A ∈ B))
5	4	biimprd 214	. . . . 5 ⊢ ((φ ∧ x = A) → (A ∈ B → x ∈ B))
6		rspcimdv.2	. . . . 5 ⊢ ((φ ∧ x = A) → (ψ → χ))
7	5, 6	imim12d 68	. . . 4 ⊢ ((φ ∧ x = A) → ((x ∈ B → ψ) → (A ∈ B → χ)))
8	2, 7	spcimdv 2937	. . 3 ⊢ (φ → (∀x(x ∈ B → ψ) → (A ∈ B → χ)))
9	2, 8	mpid 37	. 2 ⊢ (φ → (∀x(x ∈ B → ψ) → χ))
10	1, 9	syl5bi 208	1 ⊢ (φ → (∀x ∈ B ψ → χ))

Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540   = 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:  rspcimedv  2958  rspcdv  2959