Theorem sbal 1349

Description: Move universal quantifier in and out of substitution.

Assertion

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

Distinct variable groups:   x,y   x,z

Proof of Theorem sbal

Step	Hyp	Ref	Expression
1		a16gb 1279	. . . . 5 |- (A.x x = z -> (ph <-> A.xph))
2	1	sbimi 1175	. . . 4 |- ([z / y]A.x x = z -> [z / y](ph <-> A.xph))
3		sbequ5 1192	. . . 4 |- ([z / y]A.x x = z <-> A.x x = z)
4		sbbi 1241	. . . 4 |- ([z / y](ph <-> A.xph) <-> ([z / y]ph <-> [z / y]A.xph))
5	2, 3, 4	3imtr3 218	. . 3 |- (A.x x = z -> ([z / y]ph <-> [z / y]A.xph))
6		a16gb 1279	. . 3 |- (A.x x = z -> ([z / y]ph <-> A.x[z / y]ph))
7	5, 6	bitr3d 532	. 2 |- (A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))
8		sbal1 1348	. 2 |- (-. A.x x = z -> ([z / y]A.xph <-> A.x[z / y]ph))
9	7, 8	pm2.61i 126	1 |- ([z / y]A.xph <-> A.x[z / y]ph)

Syntax hints:   <-> wb 146  A.wal 956  [wsbc 1172

This theorem is referenced by:  sbex 1350  sbalv 1351  sbabel 1587  sbcalg 1977

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-12 970  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174

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