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Theorem e2ebindALT 38187
 Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. The proof is derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in e2ebindVD 38170. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
e2ebindALT (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebindALT
StepHypRef Expression
1 axc11n 2295 . 2 (∀𝑥 𝑥 = 𝑦 → ∀𝑦 𝑦 = 𝑥)
2 nfe1 2014 . . . 4 𝑦𝑦𝜑
3219.9 2060 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
4 excom 2029 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
5 nfa1 2015 . . . . . 6 𝑦𝑦 𝑦 = 𝑥
6 id 22 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → ∀𝑦 𝑦 = 𝑥)
7 biid 250 . . . . . . . . 9 (𝜑𝜑)
87a1i 11 . . . . . . . 8 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
98drex1 2315 . . . . . . 7 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
106, 9syl 17 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
115, 10alrimi 2069 . . . . 5 (∀𝑦 𝑦 = 𝑥 → ∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑))
12 exbi 1762 . . . . 5 (∀𝑦(∃𝑦𝜑 ↔ ∃𝑥𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
1311, 12syl 17 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
14 bitr 741 . . . . . 6 (((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) ∧ (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
1514ex 449 . . . . 5 ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑) → ((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)))
1615impcom 445 . . . 4 (((∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑) ∧ (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑)) → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
174, 13, 16sylancr 694 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
18 bitr3 341 . . . 4 ((∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑) → ((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑)))
1918impcom 445 . . 3 (((∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑) ∧ (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑)) → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
203, 17, 19sylancr 694 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
211, 20syl 17 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473  ∃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 This theorem is referenced by: (None)
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