Theorem spcimdv 3532

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-10 2137 and ax-11 2154. (Revised by Gino Giotto, 20-Aug-2023.)

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.1	. . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
2		elisset 2820	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
4		spcimdv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
5	4	ex 413	. . . 4 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
6	5	eximdv 1920	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒)))
7	3, 6	mpd 15	. 2 ⊢ (𝜑 → ∃𝑥(𝜓 → 𝜒))
8		19.36v 1991	. 2 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒))
9	7, 8	sylib 217	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 396  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-clel 2816

This theorem is referenced by:  spcdv  3533  spcimedv  3534  rspcimdv  3551  mrieqv2d  17348  mreexexlemd  17353  intabssd  41126