Metamath Proof Explorer


Theorem ceqsalt

Description: Closed theorem version of ceqsalg . (Contributed by NM, 28-Feb-2013) (Revised by Mario Carneiro, 10-Oct-2016)

Ref Expression
Assertion ceqsalt ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ 𝐴𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 ) )

Proof

Step Hyp Ref Expression
1 biimp ( ( 𝜑𝜓 ) → ( 𝜑𝜓 ) )
2 1 imim3i ( ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) → ( ( 𝑥 = 𝐴𝜑 ) → ( 𝑥 = 𝐴𝜓 ) ) )
3 2 al2imi ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) ) )
4 elisset ( 𝐴𝑉 → ∃ 𝑥 𝑥 = 𝐴 )
5 19.23t ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) ) )
6 5 biimpd ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) → ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) ) )
7 4 6 syl7 ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) → ( 𝐴𝑉𝜓 ) ) )
8 3 7 sylan9r ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) → ( 𝐴𝑉𝜓 ) ) )
9 8 com23 ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ) → ( 𝐴𝑉 → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) → 𝜓 ) ) )
10 9 3impia ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ 𝐴𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) → 𝜓 ) )
11 ceqsal1t ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ) )
12 11 3adant3 ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ 𝐴𝑉 ) → ( 𝜓 → ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ) )
13 10 12 impbid ( ( Ⅎ 𝑥 𝜓 ∧ ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ 𝐴𝑉 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜑 ) ↔ 𝜓 ) )