Theorem e2ebind 42072

Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 42072 is derived from e2ebindVD 42421. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
e2ebind	⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Proof of Theorem e2ebind

Step	Hyp	Ref	Expression
1		biidd 261	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
2	1	drex1 2441	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
3	2	drex2 2442	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑))
4		excom 2164	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
5	3, 4	bitrdi 286	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑))
6		nfe1 2149	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
7	6	19.9 2201	. . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
8	5, 7	bitr3di 285	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))
9	8	aecoms 2428	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   = wceq 1539  ∃wex 1783

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2139  ax-11 2156  ax-12 2173  ax-13 2372

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)