Theorem equs5e 2349

Description: A property related to substitution that unlike equs5 2351 does not require a distinctor antecedent. See equs5eALT 2178 for an alternate proof using ax-12 2047 but not ax13 2249. (Contributed by NM, 2-Feb-2007.) (Proof shortened by Wolf Lammen, 15-Jan-2018.)

Assertion

Ref	Expression
equs5e	|- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )

Proof of Theorem equs5e

Step	Hyp	Ref	Expression
1		nfa1 2028	. 2 |- F/ x A. x ( x = y -> E. y ph )
2		ax12 2304	. . 3 |- ( x = y -> ( A. y E. y ph -> A. x ( x = y -> E. y ph ) ) )
3		hbe1 2021	. . . 4 |- ( E. y ph -> A. y E. y ph )
4	3	19.23bi 2061	. . 3 |- ( ph -> A. y E. y ph )
5	2, 4	impel 485	. 2 |- ( ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
6	1, 5	exlimi 2086	1 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   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-12 2047  ax-13 2246

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

This theorem is referenced by:  sb4e  2362