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Theorem e2ebind 41203
Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 41203 is derived from e2ebindVD 41552. (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 265 . . . . . 6 (∀𝑦 𝑦 = 𝑥 → (𝜑𝜑))
21drex1 2464 . . . . 5 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
32drex2 2465 . . . 4 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑦𝑥𝜑))
4 excom 2169 . . . 4 (∃𝑦𝑥𝜑 ↔ ∃𝑥𝑦𝜑)
53, 4syl6bb 290 . . 3 (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝑦𝜑 ↔ ∃𝑥𝑦𝜑))
6 nfe1 2154 . . . 4 𝑦𝑦𝜑
7619.9 2206 . . 3 (∃𝑦𝑦𝜑 ↔ ∃𝑦𝜑)
85, 7bitr3di 289 . 2 (∀𝑦 𝑦 = 𝑥 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
98aecoms 2451 1 (∀𝑥 𝑥 = 𝑦 → (∃𝑥𝑦𝜑 ↔ ∃𝑦𝜑))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1536   = wceq 1538  wex 1781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-11 2161  ax-12 2178  ax-13 2391
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786
This theorem is referenced by: (None)
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