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Metamath Proof Explorer

Theorem ceqsal

Description: A representation of explicit substitution of a class for a variable, inferred from an implicit substitution hypothesis. (Contributed by NM, 18-Aug-1993) Avoid df-clab . (Revised by Wolf Lammen, 23-Jan-2025)

Ref Expression
Hypotheses ceqsal.1 ⊢ Ⅎ 𝑥 𝜓
ceqsal.2 ⊢ 𝐴 ∈ V
ceqsal.3 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
Assertion ceqsal ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 ceqsal.1 ⊢ Ⅎ 𝑥 𝜓
2 ceqsal.2 ⊢ 𝐴 ∈ V
3 ceqsal.3 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
4 1 19.23 ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
5 3 pm5.74i ⊢ ( ( 𝑥 = 𝐴 → 𝜑 ) ↔ ( 𝑥 = 𝐴 → 𝜓 ) )
6 5 albii ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) )
7 2 isseti ⊢ ∃ 𝑥 𝑥 = 𝐴
8 7 a1bi ⊢ ( 𝜓 ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
9 4 6 8 3bitr4i ⊢ ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜑 ) ↔ 𝜓 )