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Theorem e2ebind 39305
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 39305 is derived from e2ebindVD 39671. (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 2181 . . . 4 𝑦𝑦𝜑
2119.9 2226 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
3 biidd 252 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
43drex1 2475 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
54drex2 2476 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
6 excom 2196 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
75, 6syl6bb 276 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
82, 7syl5rbbr 275 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
98aecoms 2462 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wal 1627   = wceq 1629  wex 1850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1868  ax-4 1883  ax-5 1989  ax-6 2055  ax-7 2091  ax-10 2172  ax-11 2188  ax-12 2201  ax-13 2406
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1632  df-ex 1851  df-nf 1856
This theorem is referenced by: (None)
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