Theorem equs45f 2350

Description: Two ways of expressing substitution when y is not free in ph. The implication "to the left" is equs4 2290 and does not require the non-freeness hypothesis. Theorem sb56 2150 replaces the non-freeness hypothesis with a dv condition and equs5 2351 replaces it with a distinctor as antecedent. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
equs45f.1	|- F/ y ph

Assertion

Ref	Expression
equs45f	|- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Proof of Theorem equs45f

Step	Hyp	Ref	Expression
1		equs45f.1	. . . . . 6 |- F/ y ph
2	1	nf5ri 2065	. . . . 5 |- ( ph -> A. y ph )
3	2	anim2i 593	. . . 4 |- ( ( x = y /\ ph ) -> ( x = y /\ A. y ph ) )
4	3	eximi 1762	. . 3 |- ( E. x ( x = y /\ ph ) -> E. x ( x = y /\ A. y ph ) )
5		equs5a 2348	. . 3 |- ( E. x ( x = y /\ A. y ph ) -> A. x ( x = y -> ph ) )
6	4, 5	syl 17	. 2 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> ph ) )
7		equs4 2290	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
8	6, 7	impbii 199	1 |- ( E. x ( x = y /\ ph ) <-> A. x ( x = y -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   E.wex 1704   F/wnf 1708

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:  sb5f  2386