Theorem spcimdv 3547

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017.) Avoid ax-10 2178 and ax-11 2194. (Revised by GG, 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 2842	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
3	1, 2	syl 18	. . 3 ⊢ (𝜑 → ∃𝑥 𝑥 = 𝐴)
4		spcimdv.2	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
5	4	ex 418	. . . 4 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
6	5	eximdv 1950	. . 3 ⊢ (𝜑 → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒)))
7	3, 6	mpd 16	. 2 ⊢ (𝜑 → ∃𝑥(𝜓 → 𝜒))
8		19.36v 2026	. 2 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒))
9	7, 8	sylib 221	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-clel 2835

This theorem is used by:  spcdv  3548  spcimedv  3549  rspcimdv  3566  mrieqv2d  17774  mreexexlemd  17779  intabssd  44463