Theorem sb4e 2362

Description: One direction of a simplified definition of substitution that unlike sb4 2356 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 1883	. 2 |- ( [ y / x ] ph -> E. x ( x = y /\ ph ) )
2		equs5e 2349	. 2 |- ( E. x ( x = y /\ ph ) -> A. x ( x = y -> E. y ph ) )
3	1, 2	syl 17	1 |- ( [ y / x ] ph -> A. x ( x = y -> E. y 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-10 2019  ax-12 2047  ax-13 2246

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-ex 1705  df-nf 1710  df-sb 1881

This theorem is referenced by:  hbsb2e  2363