Theorem iotasbc2 38621

Description: Theorem *14.111 in [WhiteheadRussell] p. 184. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotasbc2	|- ( ( E! x ph /\ E! x ps ) -> ( [. ( iota x ph ) / y ]. [. ( iota x ps ) / z ]. ch <-> E. y E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) ) )

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

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

Proof of Theorem iotasbc2

Step	Hyp	Ref	Expression
1		iotasbc 38620	. 2 |- ( E! x ph -> ( [. ( iota x ph ) / y ]. [. ( iota x ps ) / z ]. ch <-> E. y ( A. x ( ph <-> x = y ) /\ [. ( iota x ps ) / z ]. ch ) ) )
2		iotasbc 38620	. . . . 5 |- ( E! x ps -> ( [. ( iota x ps ) / z ]. ch <-> E. z ( A. x ( ps <-> x = z ) /\ ch ) ) )
3	2	anbi2d 740	. . . 4 |- ( E! x ps -> ( ( A. x ( ph <-> x = y ) /\ [. ( iota x ps ) / z ]. ch ) <-> ( A. x ( ph <-> x = y ) /\ E. z ( A. x ( ps <-> x = z ) /\ ch ) ) ) )
4		3anass 1042	. . . . . 6 |- ( ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) <-> ( A. x ( ph <-> x = y ) /\ ( A. x ( ps <-> x = z ) /\ ch ) ) )
5	4	exbii 1774	. . . . 5 |- ( E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) <-> E. z ( A. x ( ph <-> x = y ) /\ ( A. x ( ps <-> x = z ) /\ ch ) ) )
6		19.42v 1918	. . . . 5 |- ( E. z ( A. x ( ph <-> x = y ) /\ ( A. x ( ps <-> x = z ) /\ ch ) ) <-> ( A. x ( ph <-> x = y ) /\ E. z ( A. x ( ps <-> x = z ) /\ ch ) ) )
7	5, 6	bitr2i 265	. . . 4 |- ( ( A. x ( ph <-> x = y ) /\ E. z ( A. x ( ps <-> x = z ) /\ ch ) ) <-> E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) )
8	3, 7	syl6bb 276	. . 3 |- ( E! x ps -> ( ( A. x ( ph <-> x = y ) /\ [. ( iota x ps ) / z ]. ch ) <-> E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) ) )
9	8	exbidv 1850	. 2 |- ( E! x ps -> ( E. y ( A. x ( ph <-> x = y ) /\ [. ( iota x ps ) / z ]. ch ) <-> E. y E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) ) )
10	1, 9	sylan9bb 736	1 |- ( ( E! x ph /\ E! x ps ) -> ( [. ( iota x ph ) / y ]. [. ( iota x ps ) / z ]. ch <-> E. y E. z ( A. x ( ph <-> x = y ) /\ A. x ( ps <-> x = z ) /\ ch ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   A.wal 1481   E.wex 1704   E!weu 2470   [.wsbc 3435   iotacio 5849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rex 2918  df-v 3202  df-sbc 3436  df-un 3579  df-sn 4178  df-pr 4180  df-uni 4437  df-iota 5851

This theorem is referenced by: (None)