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Metamath Proof Explorer

Theorem cleljustALT2

Description: Alternate proof of cleljust . Compared with cleljustALT , it uses nfv followed by equsexv instead of ax-5 followed by equsexhv , so it uses the idiom F/ x ph instead of ph -> A. x ph to express nonfreeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004) (Revised by Mario Carneiro, 21-Dec-2016) (Revised by BJ, 29-Dec-2020) (Proof modification is discouraged.) (New usage is discouraged.)

		Ref	Expression
	Assertion	cleljustALT2	⊢ ( 𝑥 ∈ 𝑦 ↔ ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) )

Proof

Step	Hyp	Ref	Expression
1		nfv	⊢ Ⅎ 𝑧 𝑥 ∈ 𝑦
2		elequ1	⊢ ( 𝑧 = 𝑥 → ( 𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦 ) )
3	1 2	equsexv	⊢ ( ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) ↔ 𝑥 ∈ 𝑦 )
4	3	bicomi	⊢ ( 𝑥 ∈ 𝑦 ↔ ∃ 𝑧 ( 𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦 ) )