Theorem iotavalsb 38634

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 2052	. 2 |- ( A. x ( ph <-> x = y ) -> E. y A. x ( ph <-> x = y ) )
2		df-eu 2474	. . 3 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		iotavalb 38631	. . . 4 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )
4		dfsbcq 3437	. . . . 5 |- ( y = ( iota x ph ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
5	4	eqcoms 2630	. . . 4 |- ( ( iota x ph ) = y -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
6	3, 5	syl6bi 243	. . 3 |- ( E! x ph -> ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) ) )
7	2, 6	sylbir 225	. 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 38	1 |- ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   = wceq 1483   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-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)