Theorem equs5a 2348

Description: A property related to substitution that unlike equs5 2351 does not require a distinctor antecedent. See equs5aALT 2177 for an alternate proof using ax-12 2047 but not ax13 2249. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
equs5a	|- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		nfa1 2028	. 2 |- F/ x A. x ( x = y -> ph )
2		ax12 2304	. . 3 |- ( x = y -> ( A. y ph -> A. x ( x = y -> ph ) ) )
3	2	imp 445	. 2 |- ( ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
4	1, 3	exlimi 2086	1 |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = 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:  equs45f  2350  sb4a  2357  bj-equs45fv  32752