Theorem sb19.21 1234

Description: Substitution with a variable not free in antecedent affects only the consequent.

Hypothesis

Ref	Expression
sb19.21.1	|- (ph -> A.xph)

Assertion

Ref	Expression
sb19.21	|- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))

Proof of Theorem sb19.21

Step	Hyp	Ref	Expression
1		sbim 1232	. 2 |- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
2		sb19.21.1	. . . 4 |- (ph -> A.xph)
3	2	sbf 1184	. . 3 |- ([y / x]ph <-> ph)
4	3	imbi1i 186	. 2 |- (([y / x]ph -> [y / x]ps) <-> (ph -> [y / x]ps))
5	1, 4	bitr 173	1 |- ([y / x](ph -> ps) <-> (ph -> [y / x]ps))

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

This theorem is referenced by:  sbco2d 1254  2mos 1446  nn1suc 5895

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-or 224  df-an 225  df-ex 979  df-sb 1170

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