Theorem sb5rf 2090

Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/yph

Assertion

Ref	Expression
sb5rf	|- (ph <-> E.y(y = x /\ [y / x]ph))

Proof of Theorem sb5rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . 4 |- F/yph
2	1	sbid2 2084	. . 3 |- ([x / y][y / x]ph <-> ph)
3		sb1 1651	. . 3 |- ([x / y][y / x]ph -> E.y(y = x /\ [y / x]ph))
4	2, 3	sylbir 204	. 2 |- (ph -> E.y(y = x /\ [y / x]ph))
5		stdpc7 1917	. . . 4 |- (y = x -> ([y / x]ph -> ph))
6	5	imp 418	. . 3 |- ((y = x /\ [y / x]ph) -> ph)
7	1, 6	exlimi 1803	. 2 |- (E.y(y = x /\ [y / x]ph) -> ph)
8	4, 7	impbii 180	1 |- (ph <-> E.y(y = x /\ [y / x]ph))

Syntax hints:   <-> wb 176   /\ wa 358  E.wex 1541   F/wnf 1544  [wsb 1648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 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  df-sb 1649

This theorem is referenced by:  2sb5rf  2117