Theorem sb19.21 1236

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 1234	. 2 |- ([y / x](ph -> ps) <-> ([y / x]ph -> [y / x]ps))
2		sb19.21.1	. . . 4 |- (ph -> A.xph)
3	2	sbf 1186	. . 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 954  [wsbc 1170

This theorem is referenced by:  sbco2d 1256  2mos 1448  nn1suc 5939

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-10 966  ax-12 968  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-11o 1218

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172

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