Theorem e2ebind 38779

Description: Absorption of an existential quantifier of a double existential quantifier of non-distinct variables. e2ebind 38779 is derived from e2ebindVD 39148. (Contributed by Alan Sare, 27-Nov-2014.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
e2ebind	|- ( A. x x = y -> ( E. x E. y ph <-> E. y ph ) )

Proof of Theorem e2ebind

Step	Hyp	Ref	Expression
1		nfe1 2027	. . . 4 |- F/ y E. y ph
2	1	19.9 2072	. . 3 |- ( E. y E. y ph <-> E. y ph )
3		biidd 252	. . . . . 6 |- ( A. y y = x -> ( ph <-> ph ) )
4	3	drex1 2327	. . . . 5 |- ( A. y y = x -> ( E. y ph <-> E. x ph ) )
5	4	drex2 2328	. . . 4 |- ( A. y y = x -> ( E. y E. y ph <-> E. y E. x ph ) )
6		excom 2042	. . . 4 |- ( E. y E. x ph <-> E. x E. y ph )
7	5, 6	syl6bb 276	. . 3 |- ( A. y y = x -> ( E. y E. y ph <-> E. x E. y ph ) )
8	2, 7	syl5rbbr 275	. 2 |- ( A. y y = x -> ( E. x E. y ph <-> E. y ph ) )
9	8	aecoms 2312	1 |- ( A. x x = y -> ( E. x E. y ph <-> E. y ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   E.wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710

This theorem is referenced by: (None)