Theorem 2sb5rf 2117

Description: Reversed double substitution. (Contributed by NM, 3-Feb-2005.) (Revised by Mario Carneiro, 6-Oct-2016.)

Hypotheses

Ref	Expression
2sb5rf.1	|- F/zph
2sb5rf.2	|- F/wph

Assertion

Ref	Expression
2sb5rf	|- (ph <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))

Distinct variable groups:   x,y   x,w   y,z   z,w

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

Proof of Theorem 2sb5rf

Step	Hyp	Ref	Expression
1		2sb5rf.1	. . 3 |- F/zph
2	1	sb5rf 2090	. 2 |- (ph <-> E.z(z = x /\ [z / x]ph))
3		19.42v 1905	. . . 4 |- (E.w(z = x /\ (w = y /\ [w / y][z / x]ph)) <-> (z = x /\ E.w(w = y /\ [w / y][z / x]ph)))
4		sbcom2 2114	. . . . . . 7 |- ([z / x][w / y]ph <-> [w / y][z / x]ph)
5	4	anbi2i 675	. . . . . 6 |- (((z = x /\ w = y) /\ [z / x][w / y]ph) <-> ((z = x /\ w = y) /\ [w / y][z / x]ph))
6		anass 630	. . . . . 6 |- (((z = x /\ w = y) /\ [w / y][z / x]ph) <-> (z = x /\ (w = y /\ [w / y][z / x]ph)))
7	5, 6	bitri 240	. . . . 5 |- (((z = x /\ w = y) /\ [z / x][w / y]ph) <-> (z = x /\ (w = y /\ [w / y][z / x]ph)))
8	7	exbii 1582	. . . 4 |- (E.w((z = x /\ w = y) /\ [z / x][w / y]ph) <-> E.w(z = x /\ (w = y /\ [w / y][z / x]ph)))
9		2sb5rf.2	. . . . . . 7 |- F/wph
10	9	nfsb 2109	. . . . . 6 |- F/w[z / x]ph
11	10	sb5rf 2090	. . . . 5 |- ([z / x]ph <-> E.w(w = y /\ [w / y][z / x]ph))
12	11	anbi2i 675	. . . 4 |- ((z = x /\ [z / x]ph) <-> (z = x /\ E.w(w = y /\ [w / y][z / x]ph)))
13	3, 8, 12	3bitr4ri 269	. . 3 |- ((z = x /\ [z / x]ph) <-> E.w((z = x /\ w = y) /\ [z / x][w / y]ph))
14	13	exbii 1582	. 2 |- (E.z(z = x /\ [z / x]ph) <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))
15	2, 14	bitri 240	1 |- (ph <-> E.zE.w((z = x /\ w = y) /\ [z / x][w / y]ph))

Syntax hints:   <-> wb 176   /\ wa 358  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: (None)