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 ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → 𝜓 )

Proof

Step Hyp Ref Expression
1 elissetv ⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑧 𝑧 = 𝐴 )
2 nfv ⊢ Ⅎ 𝑧 Ⅎ 𝑥 𝐴
3 nfnfc1 ⊢ Ⅎ 𝑥 Ⅎ 𝑥 𝐴
4 nfcvd ⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑥 𝑧 )
5 id ⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑥 𝐴 )
6 4 5 nfeqd ⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑥 𝑧 = 𝐴 )
7 6 nfnd ⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑥 ¬ 𝑧 = 𝐴 )
8 nfvd ⊢ ( Ⅎ 𝑥 𝐴 → Ⅎ 𝑧 ¬ 𝑥 = 𝐴 )
9 eqeq1 ⊢ ( 𝑧 = 𝑥 → ( 𝑧 = 𝐴 ↔ 𝑥 = 𝐴 ) )
10 9 notbid ⊢ ( 𝑧 = 𝑥 → ( ¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴 ) )
11 10 a1i ⊢ ( Ⅎ 𝑥 𝐴 → ( 𝑧 = 𝑥 → ( ¬ 𝑧 = 𝐴 ↔ ¬ 𝑥 = 𝐴 ) ) )
12 2 3 7 8 11 cbv2w ⊢ ( Ⅎ 𝑥 𝐴 → ( ∀ 𝑧 ¬ 𝑧 = 𝐴 ↔ ∀ 𝑥 ¬ 𝑥 = 𝐴 ) )
13 alnex ⊢ ( ∀ 𝑧 ¬ 𝑧 = 𝐴 ↔ ¬ ∃ 𝑧 𝑧 = 𝐴 )
14 alnex ⊢ ( ∀ 𝑥 ¬ 𝑥 = 𝐴 ↔ ¬ ∃ 𝑥 𝑥 = 𝐴 )
15 12 13 14 3bitr3g ⊢ ( Ⅎ 𝑥 𝐴 → ( ¬ ∃ 𝑧 𝑧 = 𝐴 ↔ ¬ ∃ 𝑥 𝑥 = 𝐴 ) )
16 15 con4bid ⊢ ( Ⅎ 𝑥 𝐴 → ( ∃ 𝑧 𝑧 = 𝐴 ↔ ∃ 𝑥 𝑥 = 𝐴 ) )
17 1 16 imbitrid ⊢ ( Ⅎ 𝑥 𝐴 → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) )
18 17 ad2antrr ⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( 𝐴 ∈ 𝑉 → ∃ 𝑥 𝑥 = 𝐴 ) )
19 18 3impia ⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → ∃ 𝑥 𝑥 = 𝐴 )
20 biimp ⊢ ( ( 𝜑 ↔ 𝜓 ) → ( 𝜑 → 𝜓 ) )
21 20 imim2i ⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝑥 = 𝐴 → ( 𝜑 → 𝜓 ) ) )
22 21 com23 ⊢ ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) → ( 𝜑 → ( 𝑥 = 𝐴 → 𝜓 ) ) )
23 22 imp ⊢ ( ( ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ 𝜑 ) → ( 𝑥 = 𝐴 → 𝜓 ) )
24 23 alanimi ⊢ ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) )
25 19.23t ⊢ ( Ⅎ 𝑥 𝜓 → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
26 25 adantl ⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ∀ 𝑥 ( 𝑥 = 𝐴 → 𝜓 ) ↔ ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
27 24 26 imbitrid ⊢ ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) → ( ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) ) )
28 27 imp ⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
29 28 3adant3 ⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → ( ∃ 𝑥 𝑥 = 𝐴 → 𝜓 ) )
30 19 29 mpd ⊢ ( ( ( Ⅎ 𝑥 𝐴 ∧ Ⅎ 𝑥 𝜓 ) ∧ ( ∀ 𝑥 ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) ) ∧ ∀ 𝑥 𝜑 ) ∧ 𝐴 ∈ 𝑉 ) → 𝜓 )