Theorem bj-ssb1 32633

Description: A simplified definition of substitution in case of disjoint variables. See bj-ssb1a 32632 for the backward implication, which does not require ax-11 2034 (note that here, the version of ax-11 2034 with disjoint setvar metavariables would suffice). Compare sb6 2429. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssb1	|- ([ t/ x]b ph <-> A. x ( x = t -> ph ) )

Distinct variable group:   x, t

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ssb1

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.21v 1868	. . 3 |- ( A. x ( y = t -> ( x = y -> ph ) ) <-> ( y = t -> A. x ( x = y -> ph ) ) )
2	1	albii 1747	. 2 |- ( A. y A. x ( y = t -> ( x = y -> ph ) ) <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
3		19.23v 1902	. . . . 5 |- ( A. y ( y = t -> ( x = t -> ph ) ) <-> ( E. y y = t -> ( x = t -> ph ) ) )
4		equequ2 1953	. . . . . . . 8 |- ( y = t -> ( x = y <-> x = t ) )
5	4	imbi1d 331	. . . . . . 7 |- ( y = t -> ( ( x = y -> ph ) <-> ( x = t -> ph ) ) )
6	5	pm5.74i 260	. . . . . 6 |- ( ( y = t -> ( x = y -> ph ) ) <-> ( y = t -> ( x = t -> ph ) ) )
7	6	albii 1747	. . . . 5 |- ( A. y ( y = t -> ( x = y -> ph ) ) <-> A. y ( y = t -> ( x = t -> ph ) ) )
8		ax6ev 1890	. . . . . 6 |- E. y y = t
9	8	a1bi 352	. . . . 5 |- ( ( x = t -> ph ) <-> ( E. y y = t -> ( x = t -> ph ) ) )
10	3, 7, 9	3bitr4ri 293	. . . 4 |- ( ( x = t -> ph ) <-> A. y ( y = t -> ( x = y -> ph ) ) )
11	10	albii 1747	. . 3 |- ( A. x ( x = t -> ph ) <-> A. x A. y ( y = t -> ( x = y -> ph ) ) )
12		alcom 2037	. . 3 |- ( A. x A. y ( y = t -> ( x = y -> ph ) ) <-> A. y A. x ( y = t -> ( x = y -> ph ) ) )
13	11, 12	bitri 264	. 2 |- ( A. x ( x = t -> ph ) <-> A. y A. x ( y = t -> ( x = y -> ph ) ) )
14		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
15	2, 13, 14	3bitr4ri 293	1 |- ([ t/ x]b ph <-> A. x ( x = t -> ph ) )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481   E.wex 1704  [wssb 32619

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-11 2034

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-ssb 32620

This theorem is referenced by:  bj-ax12ssb  32635  bj-ssbssblem  32649  bj-ssbcom3lem  32650