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 ( 𝑥𝑦 ↔ ∃ 𝑧 ( 𝑧 = 𝑥𝑧𝑦 ) )

Proof

Step Hyp Ref Expression
1 nfv 𝑧 𝑥𝑦
2 elequ1 ( 𝑧 = 𝑥 → ( 𝑧𝑦𝑥𝑦 ) )
3 1 2 equsexv ( ∃ 𝑧 ( 𝑧 = 𝑥𝑧𝑦 ) ↔ 𝑥𝑦 )
4 3 bicomi ( 𝑥𝑦 ↔ ∃ 𝑧 ( 𝑧 = 𝑥𝑧𝑦 ) )