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 ) )