Theorem sbi2 1231

Description: Introduction of implication into substitution.

Assertion

Ref	Expression
sbi2	|- (([y / x]ph -> [y / x]ps) -> [y / x](ph -> ps))

Proof of Theorem sbi2

Step	Hyp	Ref	Expression
1		sbn 1229	. . 3 |- ([y / x] -. ph <-> -. [y / x]ph)
2		pm2.21 76	. . . 4 |- (-. ph -> (ph -> ps))
3	2	sbimi 1171	. . 3 |- ([y / x] -. ph -> [y / x](ph -> ps))
4	1, 3	sylbir 201	. 2 |- (-. [y / x]ph -> [y / x](ph -> ps))
5		ax-1 4	. . 3 |- (ps -> (ph -> ps))
6	5	sbimi 1171	. 2 |- ([y / x]ps -> [y / x](ph -> ps))
7	4, 6	ja 137	1 |- (([y / x]ph -> [y / x]ps) -> [y / x](ph -> ps))

Syntax hints:  -. wn 2   -> wi 3  [wsbc 1168

This theorem is referenced by:  sbim 1232

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-10 964  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-an 225  df-ex 979  df-sb 1170

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