Theorem bj-spcimdvv 37986

Description: Remove from spcimdv 3551 dependency on ax-7 2043, ax-8 2149, ax-10 2180, ax-11 2197, ax-12 2216 ax-13 2404, ax-ext 2735, df-cleq 2755, df-clab 2742 (and df-nfc 2913, df-v 3456, df-or 862, df-tru 1573, df-nf 1817) at the price of adding a disjoint variable condition on 𝑥, 𝐵 (but in usages, 𝑥 is typically a dummy, hence fresh, variable). For the version without this disjoint variable condition, see bj-spcimdv 37985. (Contributed by BJ, 3-Nov-2021.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

Ref	Expression
bj-spcimdvv	⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

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

Allowed substitution hint:   𝜓(𝑥)

Proof of Theorem bj-spcimdvv

Step	Hyp	Ref	Expression
1		bj-spcimdvv.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
2	1	ex 418	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
3	2	alrimiv 1962	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)))
4		bj-spcimdvv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		elissetv 2845	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
6		exim 1867	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒)))
7	5, 6	syl5 35	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → ∃𝑥(𝜓 → 𝜒)))
8		19.36v 2028	. . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒))
9	7, 8	imbitrdi 254	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒)))
10	3, 4, 9	sylc 66	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2147

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 1945  ax-6 2002  ax-7 2043  ax-8 2149

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2839

This theorem is used by: (None)