Theorem equs45f 1989

Description: Two ways of expressing substitution when y is not free in ph. (Contributed by NM, 25-Apr-2008.) (Revised by Mario Carneiro, 4-Oct-2016.)

Hypothesis

Ref	Expression
equs45f.1	|- F/yph

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/yph
2	1	nfri 1762	. . . . 5 |- (ph -> A.yph)
3	2	anim2i 552	. . . 4 |- ((x = y /\ ph) -> (x = y /\ A.yph))
4	3	eximi 1576	. . 3 |- (E.x(x = y /\ ph) -> E.x(x = y /\ A.yph))
5		equs5a 1887	. . 3 |- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))
6	4, 5	syl 15	. 2 |- (E.x(x = y /\ ph) -> A.x(x = y -> ph))
7		equs4 1959	. 2 |- (A.x(x = y -> ph) -> E.x(x = y /\ ph))
8	6, 7	impbii 180	1 |- (E.x(x = y /\ ph) <-> A.x(x = y -> ph))

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358  A.wal 1540  E.wex 1541   F/wnf 1544

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545

This theorem is referenced by:  sb5f  2040