Theorem dfsb3 2374

Description: An alternate definition of proper substitution df-sb 1881 that uses only primitive connectives (no defined terms) on the right-hand side. (Contributed by NM, 6-Mar-2007.)

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 385	. 2 |- ( ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) <-> ( -. ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
2		dfsb2 2373	. 2 |- ( [ y / x ] ph <-> ( ( x = y /\ ph ) \/ A. x ( x = y -> ph ) ) )
3		imnan 438	. . 3 |- ( ( x = y -> -. ph ) <-> -. ( x = y /\ ph ) )
4	3	imbi1i 339	. 2 |- ( ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) <-> ( -. ( x = y /\ ph ) -> A. x ( x = y -> ph ) ) )
5	1, 2, 4	3bitr4i 292	1 |- ( [ y / x ] ph <-> ( ( x = y -> -. ph ) -> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   A.wal 1481   [wsb 1880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  sbn  2391