Theorem spcimdv

Description: Restricted specialization, using implicit substitution. (Contributed by Mario Carneiro, 4-Jan-2017) Avoid ax-10 and ax-11 . (Revised by Gino Giotto, 20-Aug-2023)

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

Proof

Step	Hyp	Ref	Expression
1		spcimdv.1	⊢ ( 𝜑 → 𝐴 ∈ 𝐵 )
2		spcimdv.2	⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 → 𝜒 ) )
3		elisset	⊢ ( 𝐴 ∈ 𝐵 → ∃ 𝑥 𝑥 = 𝐴 )
4	1 3	syl	⊢ ( 𝜑 → ∃ 𝑥 𝑥 = 𝐴 )
5	2	ex	⊢ ( 𝜑 → ( 𝑥 = 𝐴 → ( 𝜓 → 𝜒 ) ) )
6	5	eximdv	⊢ ( 𝜑 → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( 𝜓 → 𝜒 ) ) )
7	4 6	mpd	⊢ ( 𝜑 → ∃ 𝑥 ( 𝜓 → 𝜒 ) )
8		19.36v	⊢ ( ∃ 𝑥 ( 𝜓 → 𝜒 ) ↔ ( ∀ 𝑥 𝜓 → 𝜒 ) )
9	7 8	sylib	⊢ ( 𝜑 → ( ∀ 𝑥 𝜓 → 𝜒 ) )