Theorem pm14.123b 16390

Description: Theorem *14.123 in [WhiteheadRussell] p. 185.

Assertion

Ref	Expression
pm14.123b	|- ((A e. C /\ B e. D) -> ((A.zA.w(ph -> (z = A /\ w = B)) /\ [A / z][B / w]ph) <-> (A.zA.w(ph -> (z = A /\ w = B)) /\ E.zE.wph)))

Distinct variable groups:   w,A,z   w,B,z

Proof of Theorem pm14.123b

Step	Hyp	Ref	Expression
1		2sbc5g 16380	. . . . 5 |- ((A e. C /\ B e. D) -> (E.zE.w((z = A /\ w = B) /\ ph) <-> [A / z][B / w]ph))
2	1	adantr 425	. . . 4 |- (((A e. C /\ B e. D) /\ A.zA.w(ph -> (z = A /\ w = B))) -> (E.zE.w((z = A /\ w = B) /\ ph) <-> [A / z][B / w]ph))
3		hba1 1350	. . . . . 6 |- (A.zA.w(ph -> (z = A /\ w = B)) -> A.zA.zA.w(ph -> (z = A /\ w = B)))
4		hba2 1360	. . . . . . 7 |- (A.zA.w(ph -> (z = A /\ w = B)) -> A.wA.zA.w(ph -> (z = A /\ w = B)))
5		simpr 350	. . . . . . . 8 |- (((z = A /\ w = B) /\ ph) -> ph)
6		ax-4 1319	. . . . . . . . . 10 |- (A.w(ph -> (z = A /\ w = B)) -> (ph -> (z = A /\ w = B)))
7	6	a4s 1330	. . . . . . . . 9 |- (A.zA.w(ph -> (z = A /\ w = B)) -> (ph -> (z = A /\ w = B)))
8	7	ancrd 323	. . . . . . . 8 |- (A.zA.w(ph -> (z = A /\ w = B)) -> (ph -> ((z = A /\ w = B) /\ ph)))
9	5, 8	impbid2 576	. . . . . . 7 |- (A.zA.w(ph -> (z = A /\ w = B)) -> (((z = A /\ w = B) /\ ph) <-> ph))
10	4, 9	exbid 1460	. . . . . 6 |- (A.zA.w(ph -> (z = A /\ w = B)) -> (E.w((z = A /\ w = B) /\ ph) <-> E.wph))
11	3, 10	exbid 1460	. . . . 5 |- (A.zA.w(ph -> (z = A /\ w = B)) -> (E.zE.w((z = A /\ w = B) /\ ph) <-> E.zE.wph))
12	11	adantl 424	. . . 4 |- (((A e. C /\ B e. D) /\ A.zA.w(ph -> (z = A /\ w = B))) -> (E.zE.w((z = A /\ w = B) /\ ph) <-> E.zE.wph))
13	2, 12	bitr3d 589	. . 3 |- (((A e. C /\ B e. D) /\ A.zA.w(ph -> (z = A /\ w = B))) -> ([A / z][B / w]ph <-> E.zE.wph))
14	13	ex 402	. 2 |- ((A e. C /\ B e. D) -> (A.zA.w(ph -> (z = A /\ w = B)) -> ([A / z][B / w]ph <-> E.zE.wph)))
15	14	pm5.32d 709	1 |- ((A e. C /\ B e. D) -> ((A.zA.w(ph -> (z = A /\ w = B)) /\ [A / z][B / w]ph) <-> (A.zA.w(ph -> (z = A /\ w = B)) /\ E.zE.wph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534

This theorem is referenced by:  pm14.123c 16391

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-v 2294  df-sbc 2454

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