Theorem sb5rf 2422

Description: Reversed substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.) (Proof shortened by Wolf Lammen, 20-Sep-2018.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/ y ph

Assertion

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

Proof of Theorem sb5rf

Step	Hyp	Ref	Expression
1		sb5rf.1	. . 3 |- F/ y ph
2		sbequ12r 2112	. . 3 |- ( y = x -> ( [ y / x ] ph <-> ph ) )
3	1, 2	equsex 2292	. 2 |- ( E. y ( y = x /\ [ y / x ] ph ) <-> ph )
4	3	bicomi 214	1 |- ( ph <-> E. y ( y = x /\ [ y / x ] ph ) )

Syntax hints:   <-> wb 196   /\ wa 384   E.wex 1704   F/wnf 1708   [wsb 1880

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-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  df-sb 1881

This theorem is referenced by: (None)