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Metamath Proof Explorer

Theorem spcimdv

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