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Theorem e2ebindVD 38628
Description: The following User's Proof is a Virtual Deduction proof (see wvd1 38264) completed automatically by a Metamath tools program invoking mmj2 and the Metamath Proof Assistant. e2ebind 38258 is e2ebindVD 38628 without virtual deductions and was automatically derived from e2ebindVD 38628.
1:: (𝜑𝜑)
2:1: (∀𝑦𝑦 = 𝑥 → (𝜑𝜑))
3:2: (∀𝑦𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑 ))
4:: (   𝑦𝑦 = 𝑥   ▶   𝑦𝑦 = 𝑥   )
5:3,4: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥 𝜑)   )
6:: (∀𝑦𝑦 = 𝑥 → ∀𝑦𝑦𝑦 = 𝑥)
7:5,6: (   𝑦𝑦 = 𝑥   ▶   𝑦(∃𝑦𝜑 𝑥𝜑)   )
8:7: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 𝑦𝑥𝜑)   )
9:: (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
10:8,9: (   𝑦𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 𝑥𝑦𝜑)   )
11:: (∃𝑦𝜑 → ∀𝑦𝑦𝜑)
12:11: (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
13:10,12: (   𝑦𝑦 = 𝑥   ▶   (∃𝑥𝑦𝜑 𝑦𝜑)   )
14:13: (∀𝑦𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃ 𝑦𝜑))
15:: (∀𝑦𝑦 = 𝑥 ↔ ∀𝑥𝑥 = 𝑦)
qed:14,15: (∀𝑥𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃ 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 2306 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 hba1 2148 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ∀𝑦𝑦 𝑦 = 𝑥)
3 idn1 38269 . . . . . . . 8 (   𝑦 𝑦 = 𝑥   ▶   𝑦 𝑦 = 𝑥   )
4 biid 251 . . . . . . . . . 10 (𝜑𝜑)
54a1i 11 . . . . . . . . 9 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
65drex1 2326 . . . . . . . 8 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
73, 6e1a 38331 . . . . . . 7 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝜑 ↔ ∃𝑥𝜑)   )
82, 7gen11nv 38321 . . . . . 6 (   𝑦 𝑦 = 𝑥   ▶   𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑)   )
9 exbi 1770 . . . . . 6 (∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
108, 9e1a 38331 . . . . 5 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑)   )
11 excom 2039 . . . . 5 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
12 bibi1 341 . . . . . 6 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) ↔ (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)))
1312biimprd 238 . . . . 5 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)))
1410, 11, 13e10 38398 . . . 4 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)   )
15 nfe1 2024 . . . . 5 𝑦𝑦𝜑
161519.9 2070 . . . 4 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
17 bitr3 342 . . . 4 ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)))
1814, 16, 17e10 38398 . . 3 (   𝑦 𝑦 = 𝑥   ▶   (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)   )
1918in1 38266 . 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 196  wal 1478   = wceq 1480  wex 1701
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1702  df-nf 1707  df-vd1 38265
This theorem is referenced by: (None)
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