Theorem sbco2d 1254

Description: A composition law for substitution.

Hypotheses

Ref	Expression
sbco2d.1	|- (ph -> A.xph)
sbco2d.2	|- (ph -> A.zph)
sbco2d.3	|- (ph -> (ps -> A.zps))

Assertion

Ref	Expression
sbco2d	|- (ph -> ([y / z][z / x]ps <-> [y / x]ps))

Proof of Theorem sbco2d

Step	Hyp	Ref	Expression
1		sbco2d.2	. . . . 5 |- (ph -> A.zph)
2		sbco2d.3	. . . . 5 |- (ph -> (ps -> A.zps))
3	1, 2	hbim1 1101	. . . 4 |- ((ph -> ps) -> A.z(ph -> ps))
4	3	sbco2 1253	. . 3 |- ([y / z][z / x](ph -> ps) <-> [y / x](ph -> ps))
5		sbco2d.1	. . . . . 6 |- (ph -> A.xph)
6	5	sb19.21 1234	. . . . 5 |- ([z / x](ph -> ps) <-> (ph -> [z / x]ps))
7	6	sbbii 1172	. . . 4 |- ([y / z][z / x](ph -> ps) <-> [y / z](ph -> [z / x]ps))
8	1	sb19.21 1234	. . . 4 |- ([y / z](ph -> [z / x]ps) <-> (ph -> [y / z][z / x]ps))
9	7, 8	bitr 173	. . 3 |- ([y / z][z / x](ph -> ps) <-> (ph -> [y / z][z / x]ps))
10	5	sb19.21 1234	. . 3 |- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))
11	4, 9, 10	3bitr3 181	. 2 |- ((ph -> [y / z][z / x]ps) <-> (ph -> [y / x]ps))
12	11	pm5.74ri 586	1 |- (ph -> ([y / z][z / x]ps <-> [y / x]ps))

Syntax hints:   -> wi 3   <-> wb 146  A.wal 952  [wsbc 1168

This theorem is referenced by:  sbco3 1255

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-11o 1216

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170

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