Theorem 2mos 1441

Description: Double "exists at most one" with implicit substitution.

Hypothesis

Ref	Expression
2mos.1	|- ((x = z /\ y = w) -> (ph <-> ps))

Assertion

Ref	Expression
2mos	|- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))

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

Proof of Theorem 2mos

Step	Hyp	Ref	Expression
1		2mo 1440	. 2 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)))
2		ax-17 968	. . . . . . 7 |- (ps -> A.xps)
3		ax-17 968	. . . . . . . . . 10 |- (x = z -> A.y x = z)
4	3	sb19.21 1231	. . . . . . . . 9 |- ([w / y](x = z -> ph) <-> (x = z -> [w / y]ph))
5		ax-17 968	. . . . . . . . . 10 |- ((x = z -> ps) -> A.y(x = z -> ps))
6		2mos.1	. . . . . . . . . . . 12 |- ((x = z /\ y = w) -> (ph <-> ps))
7	6	expcom 374	. . . . . . . . . . 11 |- (y = w -> (x = z -> (ph <-> ps)))
8	7	pm5.74d 583	. . . . . . . . . 10 |- (y = w -> ((x = z -> ph) <-> (x = z -> ps)))
9	5, 8	sbie 1192	. . . . . . . . 9 |- ([w / y](x = z -> ph) <-> (x = z -> ps))
10	4, 9	bitr3 175	. . . . . . . 8 |- ((x = z -> [w / y]ph) <-> (x = z -> ps))
11	10	pm5.74ri 585	. . . . . . 7 |- (x = z -> ([w / y]ph <-> ps))
12	2, 11	sbie 1192	. . . . . 6 |- ([z / x][w / y]ph <-> ps)
13	12	anbi2i 479	. . . . 5 |- ((ph /\ [z / x][w / y]ph) <-> (ph /\ ps))
14	13	imbi1i 186	. . . 4 |- (((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> ((ph /\ ps) -> (x = z /\ y = w)))
15	14	2albii 997	. . 3 |- (A.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.zA.w((ph /\ ps) -> (x = z /\ y = w)))
16	15	2albii 997	. 2 |- (A.xA.yA.zA.w((ph /\ [z / x][w / y]ph) -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))
17	1, 16	bitr 173	1 |- (E.zE.wA.xA.y(ph -> (x = z /\ y = w)) <-> A.xA.yA.zA.w((ph /\ ps) -> (x = z /\ y = w)))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953  E.wex 977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168

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