Theorem rspcimdv 2625

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 2280	. 2 ⊢ (∀x ∈ B ψ ↔ ∀x(x ∈ B → ψ))
2		rspcimdv.1	. . 3 ⊢ (φ → A ∈ B)
3		simpr 103	. . . . . . 7 ⊢ ((φ ∧ x = A) → x = A)
4	3	eleq1d 2079	. . . . . 6 ⊢ ((φ ∧ x = A) → (x ∈ B ↔ A ∈ B))
5	4	biimprd 147	. . . . 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 2605	. . 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 141	1 ⊢ (φ → (∀x ∈ B ψ → χ))

Syntax hints:   → wi 4   ∧ wa 97  ∀wal 1221   = wceq 1223   ∈ wcel 1366  ∀wral 2275

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995

This theorem depends on definitions:  df-bi 110  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-ral 2280  df-v 2528

This theorem is referenced by:  rspcdv  2627