Theorem sbmo 2234

Description: Substitution into "at most one". (Contributed by Jeff Madsen, 2-Sep-2009.)

Assertion

Ref	Expression
sbmo	|- ([y / x]E*zph <-> E*z[y / x]ph)

Distinct variable groups:   x,z   y,z

Allowed substitution hints:   ph(x,y,z)

Proof of Theorem sbmo

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbex 2128	. . 3 |- ([y / x]E.wA.z(ph -> z = w) <-> E.w[y / x]A.z(ph -> z = w))
2		nfv 1619	. . . . . 6 |- F/x z = w
3	2	sblim 2068	. . . . 5 |- ([y / x](ph -> z = w) <-> ([y / x]ph -> z = w))
4	3	sbalv 2129	. . . 4 |- ([y / x]A.z(ph -> z = w) <-> A.z([y / x]ph -> z = w))
5	4	exbii 1582	. . 3 |- (E.w[y / x]A.z(ph -> z = w) <-> E.wA.z([y / x]ph -> z = w))
6	1, 5	bitri 240	. 2 |- ([y / x]E.wA.z(ph -> z = w) <-> E.wA.z([y / x]ph -> z = w))
7		nfv 1619	. . . 4 |- F/wph
8	7	mo2 2233	. . 3 |- (E*zph <-> E.wA.z(ph -> z = w))
9	8	sbbii 1653	. 2 |- ([y / x]E*zph <-> [y / x]E.wA.z(ph -> z = w))
10		nfv 1619	. . 3 |- F/w[y / x]ph
11	10	mo2 2233	. 2 |- (E*z[y / x]ph <-> E.wA.z([y / x]ph -> z = w))
12	6, 9, 11	3bitr4i 268	1 |- ([y / x]E*zph <-> E*z[y / x]ph)

Syntax hints:   -> wi 4   <-> wb 176  A.wal 1540  E.wex 1541   = wceq 1642  [wsb 1648  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)