Theorem sb4e 1829

Description: One direction of a simplified definition of substitution that unlike sb4 1856 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 1790	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		equs5e 1819	. 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 104   A.wal 1371   E.wex 1516   [wsb 1786

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-5 1471  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-11 1530  ax-4 1534  ax-ial 1558

This theorem depends on definitions:  df-bi 117  df-sb 1787

This theorem is referenced by:  hbsb2e  1831