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Metamath Proof Explorer

Theorem equs5av

Description: A property related to substitution that replaces the distinctor from equs5 to a disjoint variable condition. Version of equs5a with a disjoint variable condition, which does not require ax-13 . See also sbalex . (Contributed by NM, 2-Feb-2007) (Revised by GG, 15-Dec-2023)

		Ref	Expression
	Assertion	equs5av	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )

Proof

Step	Hyp	Ref	Expression
1		nfa1	⊢ Ⅎ 𝑥 ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 )
2		ax12v2	⊢ ( 𝑥 = 𝑦 → ( 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
3	2	spsd	⊢ ( 𝑥 = 𝑦 → ( ∀ 𝑦 𝜑 → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) ) )
4	3	imp	⊢ ( ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )
5	1 4	exlimi	⊢ ( ∃ 𝑥 ( 𝑥 = 𝑦 ∧ ∀ 𝑦 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝑦 → 𝜑 ) )