Theorem iotavalsb 36854

Description: Theorem *14.242 in [WhiteheadRussell] p. 192. (Contributed by Andrew Salmon, 11-Jul-2011.)

Assertion

Ref	Expression
iotavalsb	|- ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )

Distinct variable groups:   x, y   ph, y

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

Proof of Theorem iotavalsb

Step	Hyp	Ref	Expression
1		19.8a 1955	. 2 |- ( A. x ( ph <-> x = y ) -> E. y A. x ( ph <-> x = y ) )
2		df-eu 2323	. . 3 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		iotavalb 36851	. . . 4 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )
4		dfsbcq 3257	. . . . 5 |- ( y = ( iota x ph ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
5	4	eqcoms 2479	. . . 4 |- ( ( iota x ph ) = y -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
6	3, 5	syl6bi 236	. . 3 |- ( E! x ph -> ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) ) )
7	2, 6	sylbir 218	. 2 |- ( E. y A. x ( ph <-> x = y ) -> ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) ) )
8	1, 7	mpcom 36	1 |- ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 189   A.wal 1450   = wceq 1452   E.wex 1671   E!weu 2319   [.wsbc 3255   iotacio 5551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-rex 2762  df-v 3033  df-sbc 3256  df-un 3395  df-sn 3960  df-pr 3962  df-uni 4191  df-iota 5553

This theorem is referenced by: (None)