Theorem bj-spcimdv 36837

Description: Remove from spcimdv 3577 dependency on ax-9 2117, ax-10 2140, ax-11 2156, ax-13 2375, ax-ext 2706, df-cleq 2726 (and df-nfc 2884, df-v 3466, df-or 848, df-tru 1542, df-nf 1783). For an even more economical version, see bj-spcimdvv 36838. (Contributed by BJ, 30-Nov-2020.) (Proof modification is discouraged.)

Hypotheses

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

Assertion

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

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

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

Proof of Theorem bj-spcimdv

Step	Hyp	Ref	Expression
1		bj-spcimdv.2	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → (𝜓 → 𝜒))
2	1	ex 412	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
3	2	alrimiv 1926	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)))
4		bj-spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		elisset 2815	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
6		exim 1833	. . . 4 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (∃𝑥 𝑥 = 𝐴 → ∃𝑥(𝜓 → 𝜒)))
7	5, 6	syl5 34	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → ∃𝑥(𝜓 → 𝜒)))
8		19.36v 1986	. . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒))
9	7, 8	imbitrdi 251	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)) → (𝐴 ∈ 𝐵 → (∀𝑥𝜓 → 𝜒)))
10	3, 4, 9	sylc 65	1 ⊢ (𝜑 → (∀𝑥𝜓 → 𝜒))

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1537   = wceq 1539  ∃wex 1778   ∈ wcel 2107

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-clel 2808

This theorem is referenced by: (None)