Theorem mojust 2542

Description: Soundness justification theorem for df-mo 2543. (Contributed by NM, 11-Mar-2010.) Added this theorem by adapting the proof of eujust 2575. (Revised by BJ, 30-Sep-2022.)

Assertion

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

Distinct variable groups:   𝑥,𝑦,𝑧   𝜑,𝑦,𝑧

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem mojust

Step	Hyp	Ref	Expression
1		equequ2 2033	. . . 4 ⊢ (𝑦 = 𝑧 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑧))
2	1	imbi2d 341	. . 3 ⊢ (𝑦 = 𝑧 → ((𝜑 → 𝑥 = 𝑦) ↔ (𝜑 → 𝑥 = 𝑧)))
3	2	albidv 1927	. 2 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∀𝑥(𝜑 → 𝑥 = 𝑧)))
4	3	cbvexvw 2044	1 ⊢ (∃𝑦∀𝑥(𝜑 → 𝑥 = 𝑦) ↔ ∃𝑧∀𝑥(𝜑 → 𝑥 = 𝑧))

Syntax hints:   → wi 4   ↔ wb 207  ∀wal 1545  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015

This theorem depends on definitions:  df-bi 208  df-an 397  df-ex 1787

This theorem is referenced by:  dfmo  2544