Theorem sb8e 2093

Description: Substitution of variable in existential quantifier. (Contributed by NM, 12-Aug-1993.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypothesis

Ref	Expression
sb5rf.1	|- F/yph

Assertion

Ref	Expression
sb8e	|- (E.xph <-> E.y[y / x]ph)

Proof of Theorem sb8e

Step	Hyp	Ref	Expression
1		sb5rf.1	. . . . . 6 |- F/yph
2	1	nfn 1793	. . . . 5 |- F/y -. ph
3	2	sb8 2092	. . . 4 |- (A.x -. ph <-> A.y[y / x] -. ph)
4		sbn 2062	. . . . 5 |- ([y / x] -. ph <-> -. [y / x]ph)
5	4	albii 1566	. . . 4 |- (A.y[y / x] -. ph <-> A.y -. [y / x]ph)
6	3, 5	bitri 240	. . 3 |- (A.x -. ph <-> A.y -. [y / x]ph)
7	6	notbii 287	. 2 |- (-. A.x -. ph <-> -. A.y -. [y / x]ph)
8		df-ex 1542	. 2 |- (E.xph <-> -. A.x -. ph)
9		df-ex 1542	. 2 |- (E.y[y / x]ph <-> -. A.y -. [y / x]ph)
10	7, 8, 9	3bitr4i 268	1 |- (E.xph <-> E.y[y / x]ph)

Syntax hints:  -. wn 3   <-> wb 176  A.wal 1540  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:  exsbOLD  2131  sb8mo  2223