Theorem ax11b 1220

Description: A bidirectional version of ax-11o 1218.

Assertion

Ref	Expression
ax11b	|- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))

Proof of Theorem ax11b

Step	Hyp	Ref	Expression
1		ax-11o 1218	. . 3 |- (-. A.x x = y -> (x = y -> (ph -> A.x(x = y -> ph))))
2	1	imp 350	. 2 |- ((-. A.x x = y /\ x = y) -> (ph -> A.x(x = y -> ph)))
3		ax-4 973	. . . 4 |- (A.x(x = y -> ph) -> (x = y -> ph))
4	3	com12 11	. . 3 |- (x = y -> (A.x(x = y -> ph) -> ph))
5	4	adantl 388	. 2 |- ((-. A.x x = y /\ x = y) -> (A.x(x = y -> ph) -> ph))
6	2, 5	impbid 516	1 |- ((-. A.x x = y /\ x = y) -> (ph <-> A.x(x = y -> ph)))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 973  ax-11o 1218

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

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