Theorem iotavalsb 36642

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 1908	. 2 |- ( A. x ( ph <-> x = y ) -> E. y A. x ( ph <-> x = y ) )
2		df-eu 2269	. . 3 |- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
3		iotavalb 36639	. . . 4 |- ( E! x ph -> ( A. x ( ph <-> x = y ) <-> ( iota x ph ) = y ) )
4		dfsbcq 3301	. . . . 5 |- ( y = ( iota x ph ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
5	4	eqcoms 2434	. . . 4 |- ( ( iota x ph ) = y -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )
6	3, 5	syl6bi 231	. . 3 |- ( E! x ph -> ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) ) )
7	2, 6	sylbir 216	. 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 37	1 |- ( A. x ( ph <-> x = y ) -> ( [. y / z ]. ps <-> [. ( iota x ph ) / z ]. ps ) )

Syntax hints:   -> wi 4   <-> wb 187   A.wal 1435   = wceq 1437   E.wex 1659   E!weu 2265   [.wsbc 3299   iotacio 5560

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-rex 2781  df-v 3083  df-sbc 3300  df-un 3441  df-sn 3997  df-pr 3999  df-uni 4217  df-iota 5562

This theorem is referenced by: (None)