Theorem bj-spcimdvv

Description: Remove from spcimdv dependency on ax-7 , ax-8 , ax-10 , ax-11 , ax-12 ax-13 , ax-ext , df-cleq , df-clab (and df-nfc , df-v , df-or , df-tru , df-nf ) 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 . (Contributed by BJ, 3-Nov-2021) (Proof modification is discouraged.)

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

Proof

Step	Hyp	Ref	Expression
1		bj-spcimdvv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		bj-spcimdvv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) )
3	2	ex	⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) )
4	3	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) )
5		elissetv	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
6		exim	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) )
7	5 6	syl5	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( 𝐴 ∈ 𝐵 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) )
8		19.36v	⊢ ( ∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 𝜓 → 𝜒 ) )
9	7 8	syl6ib	⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) → ( 𝐴 ∈ 𝐵 → ( ∀ 𝑥 𝜓 → 𝜒 ) ) )
10	4 1 9	sylc	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )