Theorem sb2 2352

Description: One direction of a simplified definition of substitution. The converse requires either a dv condition (sb6 2429) or a non-freeness hypothesis (sb6f 2385). (Contributed by NM, 13-May-1993.)

Assertion

Ref	Expression
sb2	|- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Proof of Theorem sb2

Step	Hyp	Ref	Expression
1		sp 2053	. 2 |- ( A. x ( x = y -> ph ) -> ( x = y -> ph ) )
2		equs4 2290	. 2 |- ( A. x ( x = y -> ph ) -> E. x ( x = y /\ ph ) )
3		df-sb 1881	. 2 |- ( [ y / x ] ph <-> ( ( x = y -> ph ) /\ E. x ( x = y /\ ph ) ) )
4	1, 2, 3	sylanbrc 698	1 |- ( A. x ( x = y -> ph ) -> [ y / x ] ph )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   E.wex 1704   [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-12 2047  ax-13 2246

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

This theorem is referenced by:  stdpc4  2353  sb3  2355  sb4b  2358  hbsb2  2359  hbsb2a  2361  hbsb2e  2363  equsb1  2368  equsb2  2369  dfsb2  2373  sbequi  2375  sb6f  2385  sbi1  2392  sb6  2429  iota4  5869  wl-lem-moexsb  33350  sbeqal1  38598