Theorem bj-spcimdv 34214

Description: Remove from spcimdv 3592 dependency on ax-9 2124, ax-10 2145, ax-11 2161, ax-13 2390, ax-ext 2793, df-cleq 2814 (and df-nfc 2963, df-v 3496, df-or 844, df-tru 1540, df-nf 1785). For an even more economical version, see bj-spcimdvv 34215. (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 415	. . 3 ⊢ (𝜑 → (𝑥 = 𝐴 → (𝜓 → 𝜒)))
3	2	alrimiv 1928	. 2 ⊢ (𝜑 → ∀𝑥(𝑥 = 𝐴 → (𝜓 → 𝜒)))
4		bj-spcimdv.1	. 2 ⊢ (𝜑 → 𝐴 ∈ 𝐵)
5		bj-elisset 34195	. . . 4 ⊢ (𝐴 ∈ 𝐵 → ∃𝑥 𝑥 = 𝐴)
6		exim 1834	. . . 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 1994	. . 3 ⊢ (∃𝑥(𝜓 → 𝜒) ↔ (∀𝑥𝜓 → 𝜒))
9	7, 8	syl6ib 253	. 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 398  ∀wal 1535   = wceq 1537  ∃wex 1780   ∈ wcel 2114

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1540  df-ex 1781  df-sb 2070  df-clab 2800  df-clel 2893

This theorem is referenced by: (None)