Theorem axi11e 2332

Description: Axiom of Variable Substitution for Existence (intuitionistic logic axiom ax-i11e). This can be derived from ax-11 1746 in a classical context but a separate axiom is needed for intuitionistic predicate calculus. (Contributed by Jim Kingdon, 31-Dec-2017.)

Assertion

Ref	Expression
axi11e	|- (x = y -> (E.x(x = y /\ ph) -> E.yph))

Proof of Theorem axi11e

Step	Hyp	Ref	Expression
1		ax-11 1746	. . 3 |- (x = y -> (A.y -. ph -> A.x(x = y -> -. ph)))
2		alnex 1543	. . 3 |- (A.y -. ph <-> -. E.yph)
3		alinexa 1578	. . 3 |- (A.x(x = y -> -. ph) <-> -. E.x(x = y /\ ph))
4	1, 2, 3	3imtr3g 260	. 2 |- (x = y -> (-. E.yph -> -. E.x(x = y /\ ph)))
5	4	con4d 97	1 |- (x = y -> (E.x(x = y /\ ph) -> E.yph))

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358  A.wal 1540  E.wex 1541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-11 1746

This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542

This theorem is referenced by: (None)