Theorem dfsb3 1263

Description: An alternate definition of proper substitution df-sb 1209 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 222	. 2 |- (((x = y /\ ph) \/ A.x(x = y -> ph)) <-> (-. (x = y /\ ph) -> A.x(x = y -> ph)))
2		dfsb2 1262	. 2 |- ([y / x]ph <-> ((x = y /\ ph) \/ A.x(x = y -> ph)))
3		imnan 240	. . 3 |- ((x = y -> -. ph) <-> -. (x = y /\ ph))
4	3	imbi1i 184	. 2 |- (((x = y -> -. ph) -> A.x(x = y -> ph)) <-> (-. (x = y /\ ph) -> A.x(x = y -> ph)))
5	1, 2, 4	3bitr4i 181	1 |- ([y / x]ph <-> ((x = y -> -. ph) -> A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   \/ wo 220   /\ wa 221  A.wal 990   = wceq 992  [wsbc 1207

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-10 1002  ax-12 1004  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-11o 1255

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209

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