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Theorem e2ebind 44560
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 44560 is derived from e2ebindVD 44909. (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 biidd 262 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
21drex1 2443 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
32drex2 2444 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
4 excom 2159 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
53, 4bitrdi 287 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
6 nfe1 2147 . . . 4 𝑦𝑦𝜑
7619.9 2202 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
85, 7bitr3di 286 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
98aecoms 2430 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1534   = wceq 1536  wex 1775
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-10 2138  ax-11 2154  ax-12 2174  ax-13 2374
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780
This theorem is referenced by: (None)
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