Theorem sb6rf 2091

Description: Reversed substitution. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/yph

Assertion

Ref	Expression
sb6rf	|- (ph <-> A.y(y = x -> [y / x]ph))

Proof of Theorem sb6rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . 3 |- F/yph
2		sbequ1 1918	. . . . 5 |- (x = y -> (ph -> [y / x]ph))
3	2	equcoms 1681	. . . 4 |- (y = x -> (ph -> [y / x]ph))
4	3	com12 27	. . 3 |- (ph -> (y = x -> [y / x]ph))
5	1, 4	alrimi 1765	. 2 |- (ph -> A.y(y = x -> [y / x]ph))
6		sb2 2023	. . 3 |- (A.y(y = x -> [y / x]ph) -> [x / y][y / x]ph)
7	1	sbid2 2084	. . 3 |- ([x / y][y / x]ph <-> ph)
8	6, 7	sylib 188	. 2 |- (A.y(y = x -> [y / x]ph) -> ph)
9	5, 8	impbii 180	1 |- (ph <-> A.y(y = x -> [y / x]ph))

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540   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:  2sb6rf  2118  eu1  2225