Theorem sb8mo 2223

Description: Variable substitution in uniqueness quantifier. (Contributed by Alexander van der Vekens, 17-Jun-2017.)

Hypothesis

Ref	Expression
sb8eu.1	|- F/yph

Assertion

Ref	Expression
sb8mo	|- (E*xph <-> E*y[y / x]ph)

Proof of Theorem sb8mo

Step	Hyp	Ref	Expression
1		sb8eu.1	. . . 4 |- F/yph
2	1	sb8e 2093	. . 3 |- (E.xph <-> E.y[y / x]ph)
3	1	sb8eu 2222	. . 3 |- (E!xph <-> E!y[y / x]ph)
4	2, 3	imbi12i 316	. 2 |- ((E.xph -> E!xph) <-> (E.y[y / x]ph -> E!y[y / x]ph))
5		df-mo 2209	. 2 |- (E*xph <-> (E.xph -> E!xph))
6		df-mo 2209	. 2 |- (E*y[y / x]ph <-> (E.y[y / x]ph -> E!y[y / x]ph))
7	4, 5, 6	3bitr4i 268	1 |- (E*xph <-> E*y[y / x]ph)

Syntax hints:   -> wi 4   <-> wb 176  E.wex 1541   F/wnf 1544  [wsb 1648  E!weu 2204  E*wmo 2205

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-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209

This theorem is referenced by: (None)