Theorem sb4b 1715

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 1713	. 2 |- ( -. A. x x = y -> ( [ y / x ] ph -> A. x ( x = y -> ph ) ) )
2		sb2 1650	. 2 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )
3	1, 2	impbid1 130	1 |- ( -. A. x x = y -> ( [ y / x ] ph <-> A. x ( x = y -> ph ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 98   A.wal 1241   [wsb 1645

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646

This theorem is referenced by: (None)