Theorem rspcimdv 2651

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 2305	. 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 2103	. . . . . 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 2631	. . 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 1240   = wceq 1242   ∈ wcel 1390  ∀wral 2300

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553

This theorem is referenced by:  rspcdv  2653