Theorem bj-ssbssblem 32649

Description: Composition of two substitutions with a fresh intermediate variable. Remark: does not seem useful. (Contributed by BJ, 22-Dec-2020.)

Assertion

Ref	Expression
bj-ssbssblem	|- ([ t/ y]b[ y/ x]b ph <-> [ t/ x]b ph )

Distinct variable groups:   y, t   x, y   ph, y

Allowed substitution hints:   ph( x, t)

Proof of Theorem bj-ssbssblem

Step	Hyp	Ref	Expression
1		bj-ssb1 32633	. 2 |- ([ t/ y]b A. x ( x = y -> ph ) <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
2		bj-ssb1 32633	. . 3 |- ([ y/ x]b ph <-> A. x ( x = y -> ph ) )
3	2	bj-ssbbii 32624	. 2 |- ([ t/ y]b[ y/ x]b ph <-> [ t/ y]b A. x ( x = y -> ph ) )
4		df-ssb 32620	. 2 |- ([ t/ x]b ph <-> A. y ( y = t -> A. x ( x = y -> ph ) ) )
5	1, 3, 4	3bitr4i 292	1 |- ([ t/ y]b[ y/ x]b ph <-> [ t/ x]b ph )

Syntax hints:   -> wi 4   <-> wb 196   A.wal 1481  [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: (None)