Theorem sb4b 1845

Description: Simplified definition of substitution when variables are distinct. (Contributed by NM, 27-May-1997.)

Assertion

Ref	Expression
sb4b	|- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Proof of Theorem sb4b

Step	Hyp	Ref	Expression
1		sb4 1843	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 1778	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	impbid1 142	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 105   A.wal 1362   [wsb 1773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774

This theorem is referenced by: (None)