Metamath Proof Explorer


Theorem ceqsexvOLD

Description: Obsolete version of ceqsexv as of 12-Oct-2024. (Contributed by NM, 2-Mar-1995) Avoid ax-12 . (Revised by Gino Giotto, 12-Oct-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses ceqsexvOLD.1 𝐴 ∈ V
ceqsexvOLD.2 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
Assertion ceqsexvOLD ( ∃ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 )

Proof

Step Hyp Ref Expression
1 ceqsexvOLD.1 𝐴 ∈ V
2 ceqsexvOLD.2 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) )
3 2 biimpa ( ( 𝑥 = 𝐴𝜑 ) → 𝜓 )
4 3 exlimiv ( ∃ 𝑥 ( 𝑥 = 𝐴𝜑 ) → 𝜓 )
5 2 biimprcd ( 𝜓 → ( 𝑥 = 𝐴𝜑 ) )
6 5 alrimiv ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) )
7 1 isseti 𝑥 𝑥 = 𝐴
8 exintr ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → ∃ 𝑥 ( 𝑥 = 𝐴𝜑 ) ) )
9 6 7 8 mpisyl ( 𝜓 → ∃ 𝑥 ( 𝑥 = 𝐴𝜑 ) )
10 4 9 impbii ( ∃ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 )