Metamath Proof Explorer


Theorem vtoclgft

Description: Closed theorem form of vtoclgf . The reverse implication is proven in ceqsal1t . See ceqsalt for a version with x and A disjoint. (Contributed by NM, 17-Feb-2013) (Revised by Mario Carneiro, 12-Oct-2016) (Proof shortened by JJ, 11-Aug-2021) Avoid ax-13 . (Revised by Gino Giotto, 6-Oct-2023)

Ref Expression
Assertion vtoclgft ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴𝑉 ) → 𝜓 )

Proof

Step Hyp Ref Expression
1 elex ( 𝐴𝑉𝐴 ∈ V )
2 issetft ( 𝑥 𝐴 → ( 𝐴 ∈ V ↔ ∃ 𝑥 𝑥 = 𝐴 ) )
3 1 2 imbitrid ( 𝑥 𝐴 → ( 𝐴𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) )
4 3 ad2antrr ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( 𝐴𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) )
5 4 3impia ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴𝑉 ) → ∃ 𝑥 𝑥 = 𝐴 )
6 biimp ( ( 𝜑𝜓 ) → ( 𝜑𝜓 ) )
7 6 imim2i ( ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) )
8 7 com23 ( ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) → ( 𝜑 → ( 𝑥 = 𝐴𝜓 ) ) )
9 8 imp ( ( ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ 𝜑 ) → ( 𝑥 = 𝐴𝜓 ) )
10 9 alanimi ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) )
11 19.23t ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) ) )
12 11 adantl ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) ) )
13 10 12 imbitrid ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) ) )
14 13 imp ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) )
15 14 3adant3 ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴𝑉 ) → ( ∃ 𝑥 𝑥 = 𝐴𝜓 ) )
16 5 15 mpd ( ( ( 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴𝑉 ) → 𝜓 )