Theorem dfsb3 1225

Description: An alternate definition of proper substitution df-sb 1171 that uses only primitive connectives (no defined terms) on the right-hand side.

Assertion

Ref	Expression
dfsb3	|- ([y / x]ph <-> ((x = y -> -. ph) -> A.x(x = y -> ph)))

Proof of Theorem dfsb3

Step	Hyp	Ref	Expression
1		df-or 224	. 2 |- (((x = y /\ ph) \/ A.x(x = y -> ph)) <-> (-. (x = y /\ ph) -> A.x(x = y -> ph)))
2		dfsb2 1224	. 2 |- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))
3		imnan 242	. . 3 |- ((x = y -> -. ph) <-> -. (x = y /\ ph))
4	3	imbi1i 186	. 2 |- (((x = y -> -. ph) -> A.x(x = y -> ph)) <-> (-. (x = y /\ ph) -> A.x(x = y -> ph)))
5	1, 2, 4	3bitr4 183	1 |- ([y / x]ph <-> ((x = y -> -. ph) -> A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223  A.wal 953   = wceq 955  [wsbc 1169

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-10 965  ax-12 967  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-11o 1217

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 980  df-sb 1171

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