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Metamath Proof Explorer

Theorem mojust

Description: Soundness justification theorem for df-mo (note that y and z need not be disjoint, although the weaker theorem with that disjoint variable condition added would be enough to justify the soundness of the definition). (Contributed by NM, 11-Mar-2010) Added this theorem by adapting the proof of eujust . (Revised by BJ, 30-Sep-2022)

Ref Expression
Assertion mojust ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) )

Proof

Step Hyp Ref Expression
1 equequ2 ⊢ ( 𝑦 = 𝑡 → ( 𝑥 = 𝑦 ↔ 𝑥 = 𝑡 ) )
2 1 imbi2d ⊢ ( 𝑦 = 𝑡 → ( ( 𝜑 → 𝑥 = 𝑦 ) ↔ ( 𝜑 → 𝑥 = 𝑡 ) ) )
3 2 albidv ⊢ ( 𝑦 = 𝑡 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑡 ) ) )
4 3 cbvexvw ⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∃ 𝑡 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑡 ) )
5 equequ2 ⊢ ( 𝑡 = 𝑧 → ( 𝑥 = 𝑡 ↔ 𝑥 = 𝑧 ) )
6 5 imbi2d ⊢ ( 𝑡 = 𝑧 → ( ( 𝜑 → 𝑥 = 𝑡 ) ↔ ( 𝜑 → 𝑥 = 𝑧 ) ) )
7 6 albidv ⊢ ( 𝑡 = 𝑧 → ( ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑡 ) ↔ ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) ) )
8 7 cbvexvw ⊢ ( ∃ 𝑡 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑡 ) ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) )
9 4 8 bitri ⊢ ( ∃ 𝑦 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑦 ) ↔ ∃ 𝑧 ∀ 𝑥 ( 𝜑 → 𝑥 = 𝑧 ) )