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Theorem e2ebind 39602
 Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 39602 is derived from e2ebindVD 39961. (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 2201 . . . 4 𝑦𝑦𝜑
2119.9 2247 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
3 biidd 254 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
43drex1 2462 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
54drex2 2463 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
6 excom 2212 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
75, 6syl6bb 279 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
82, 7syl5rbbr 278 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
98aecoms 2449 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1654   = wceq 1656  ∃wex 1878 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-tru 1660  df-ex 1879  df-nf 1883 This theorem is referenced by: (None)
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