Theorem sb3 860

Description: One direction of a simplified definition of substitution when variables are distinct.

Assertion

Ref	Expression
sb3	|- (-. A.x x = y -> (E.x(x = y /\ ph) -> [y / x]ph))

Proof of Theorem sb3

Step	Hyp	Ref	Expression
1		eqs5 832	. 2 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> A.x(x = y -> ph)))
2		sb2 859	. 2 |- (A.x(x = y -> ph) -> [y / x]ph)
3	1, 2	syl6 23	1 |- (-. A.x x = y -> (E.x(x = y /\ ph) -> [y / x]ph))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196  A.wal 672  E.wex 678   = weq 797  [wsb 852

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853

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