Theorem e2ebind 37800

Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 37800 is derived from e2ebindVD 38170. (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		nfe1 2014	. . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑
2	1	19.9 2060	. . 3 ⊢ (∃𝑦∃𝑦𝜑 ↔ ∃𝑦𝜑)
3		biidd 251	. . . . . 6 ⊢ (∀𝑦 𝑦 = 𝑥 → (𝜑 ↔ 𝜑))
4	3	drex1 2315	. . . . 5 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦𝜑 ↔ ∃𝑥𝜑))
5	4	drex2 2316	. . . 4 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑦∃𝑥𝜑))
6		excom 2029	. . . 4 ⊢ (∃𝑦∃𝑥𝜑 ↔ ∃𝑥∃𝑦𝜑)
7	5, 6	syl6bb 275	. . 3 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑦∃𝑦𝜑 ↔ ∃𝑥∃𝑦𝜑))
8	2, 7	syl5rbbr 274	. 2 ⊢ (∀𝑦 𝑦 = 𝑥 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))
9	8	aecoms 2300	1 ⊢ (∀𝑥 𝑥 = 𝑦 → (∃𝑥∃𝑦𝜑 ↔ ∃𝑦𝜑))

Syntax hints:   → wi 4   ↔ wb 195  ∀wal 1473   = wceq 1475  ∃wex 1695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701

This theorem is referenced by: (None)