Theorem sbal 997

Description: Move universal quantifier in and out of substitution.

Assertion

Ref	Expression
sbal	|- ([z / y]A.xph <-> A.x[z / y]ph)

Distinct variable group(s):   x,y   x,z

Proof of Theorem sbal

Step	Hyp	Ref	Expression
1		a16gb 934	. . . . 5 |- (A.x x = z -> (ph <-> A.xph))
2	1	sbimi 854	. . . 4 |- ([z / y]A.x x = z -> [z / y](ph <-> A.xph))
3		sbequ5 898	. . . 4 |- ([z / y]A.x x = z <-> A.x x = z)
4		sbbi 890	. . . 4 |- ([z / y](ph <-> A.xph) <-> ([z / y]ph <-> [z / y]A.xph))
5	2, 3, 4	3imtr3 191	. . 3 |- (A.x x = z -> ([z / y]ph <-> [z / y]A.xph))
6		a16gb 934	. . 3 |- (A.x x = z -> ([z / y]ph <-> A.x[z / y]ph))
7	5, 6	bitr3d 408	. 2 |- (A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))
8		sbal1 996	. 2 |- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))
9	7, 8	pm2.61i 110	1 |- ([z / y]A.xph <-> A.x[z / y]ph)

Syntax hints:   <-> wb 127  A.wal 672   = weq 797  [wsb 852

This theorem is referenced by:  sbex 998  sbalv 999  sbabel 1189  sbcal 1464

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853

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