Theorem rspcimdv 2912

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 2516	. 2 ⊢ (∀𝑥 ∈ 𝐵 𝜓 ↔ ∀𝑥(𝑥 ∈ 𝐵 → 𝜓))
2		rspcimdv.1	. . 3 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
3		simpr 110	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
4	3	eleq1d 2300	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝑥 ∈ 𝐵 ↔ 𝐴 ∈ 𝐵))
5	4	biimprd 158	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝐴 ∈ 𝐵 → 𝑥 ∈ 𝐵))
6		rspcimdv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
7	5, 6	imim12d 74	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → ((𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒)))
8	2, 7	spcimdv 2891	. . 3 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → (𝐴 ∈ 𝐵 → 𝜒)))
9	2, 8	mpid 42	. 2 ⊢ (𝜑 → (∀𝑥(𝑥 ∈ 𝐵 → 𝜓) → 𝜒))
10	1, 9	biimtrid 152	1 ⊢ (𝜑 → (∀𝑥 ∈ 𝐵 𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 104  ∀wal 1396   = wceq 1398   ∈ wcel 2202  ∀wral 2511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805

This theorem is referenced by:  rspcdv  2914  wrd2ind  11370  mpomulcn  15377