Theorem equs5a 1203

Description: A property related to substitution that unlike equs5 1227 doesn't require a distinctor antecedent.

Assertion

Ref	Expression
equs5a	|- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))

Proof of Theorem equs5a

Step	Hyp	Ref	Expression
1		hba1 1009	. 2 |- (A.x(x = y -> ph) -> A.xA.x(x = y -> ph))
2		ax-11 971	. . 3 |- (x = y -> (A.yph -> A.x(x = y -> ph)))
3	2	imp 350	. 2 |- ((x = y /\ A.yph) -> A.x(x = y -> ph))
4	1, 3	19.23ai 1070	1 |- (E.x(x = y /\ A.yph) -> A.x(x = y -> ph))

Syntax hints:   -> wi 3   /\ wa 223  A.wal 958   = wceq 960  E.wex 984

This theorem is referenced by:  sb4a 1205  equs45f 1206

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 967  ax-11 971  ax-4 977  ax-5o 979  ax-6o 982

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 985

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