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Theorem e2ebind 38261
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 38261 is derived from e2ebindVD 38631. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
e2ebind (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebind
StepHypRef Expression
1 nfe1 2024 . . . 4 𝑦𝑦𝜑
2119.9 2070 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
3 biidd 252 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
43drex1 2326 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
54drex2 2327 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
6 excom 2039 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
75, 6syl6bb 276 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
82, 7syl5rbbr 275 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
98aecoms 2311 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-tru 1483  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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