Theorem sb4e 1686

Description: One direction of a simplified definition of substitution that unlike sb4 1713 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)

Assertion

Ref	Expression
sb4e	|- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )

Proof of Theorem sb4e

Step	Hyp	Ref	Expression
1		sb1 1649	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		equs5e 1676	. 2 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
3	1, 2	syl 14	1 |- ( [ y / x ] ph -> A. x ( x = y -> E. y ph ) )

Syntax hints:   -> wi 4   /\ wa 97   A.wal 1241   E.wex 1381   [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-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-11 1397  ax-4 1400  ax-ial 1427

This theorem depends on definitions:  df-bi 110  df-sb 1646

This theorem is referenced by:  hbsb2e  1688