Theorem spcimdv 3290

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

Hypotheses

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

Assertion

Ref	Expression
spcimdv	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

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

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

Proof of Theorem spcimdv

Step	Hyp	Ref	Expression
1		spcimdv.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
2	1	ex 450	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
3	2	alrimiv 1855	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)))
4		spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		nfv 1843	. . 3 ⊢ Ⅎ𝑥𝜒
6		nfcv 2764	. . 3 ⊢ Ⅎ𝑥𝐴
7	5, 6	spcimgft 3284	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒)))
8	3, 4, 7	sylc 65	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202

This theorem is referenced by:  spcdv  3291  spcimedv  3292  rspcimdv  3310  mrieqv2d  16299  mreexexlemd  16304  intabssd  37916