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Theorem e2ebindVD 38170
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 37806) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. e2ebind 37800 is e2ebindVD 38170 without virtual deductions and was automatically derived from e2ebindVD 38170.
 1:: ⊢ (𝜑 ↔ 𝜑) 2:1: ⊢ (∀𝑦𝑦 = 𝑥 → (𝜑 ↔ 𝜑)) 3:2: ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑 )) 4:: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   ∀𝑦𝑦 = 𝑥   ) 5:3,4: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥 𝜑)   ) 6:: ⊢ (∀𝑦𝑦 = 𝑥 → ∀𝑦∀𝑦𝑦 = 𝑥) 7:5,6: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   ∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑)   ) 8:7: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑)   ) 9:: ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑) 10:8,9: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑)   ) 11:: ⊢ (∃𝑦𝜑 → ∀𝑦∃𝑦𝜑) 12:11: ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑) 13:10,12: ⊢ (   ∀𝑦𝑦 = 𝑥   ▶   (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑)   ) 14:13: ⊢ (∀𝑦𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃ 𝑦𝜑)) 15:: ⊢ (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦) qed:14,15: ⊢ (∀𝑥𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃ 𝑦𝜑))
(Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
e2ebindVD (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebindVD
StepHypRef Expression
1 axc11n 2295 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 hba1 2137 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ∀𝑦𝑦 𝑦 = 𝑥)
3 idn1 37811 . . . . . . . 8 (   𝑦 𝑦 = 𝑥   ▶   𝑦 𝑦 = 𝑥   )
4 biid 250 . . . . . . . . . 10 (𝜑𝜑)
54a1i 11 . . . . . . . . 9 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
65drex1 2315 . . . . . . . 8 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
73, 6e1a 37873 . . . . . . 7 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥𝜑)   )
82, 7gen11nv 37863 . . . . . 6 (   𝑦 𝑦 = 𝑥   ▶   𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑)   )
9 exbi 1762 . . . . . 6 (∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
108, 9e1a 37873 . . . . 5 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑)   )
11 excom 2029 . . . . 5 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
12 bibi1 340 . . . . . 6 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) ↔ (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)))
1312biimprd 237 . . . . 5 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)))
1410, 11, 13e10 37940 . . . 4 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)   )
15 nfe1 2014 . . . . 5 𝑦𝑦𝜑
161519.9 2060 . . . 4 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
17 bitr3 341 . . . 4 ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)))
1814, 16, 17e10 37940 . . 3 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)   )
1918in1 37808 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
201, 19syl 17 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-vd1 37807 This theorem is referenced by: (None)
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