Theorem bnj79OLD 12441

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj79OLD	|- ([y / x]z Fn x <-> z Fn y)

Distinct variable groups:   x,z   y,z

Proof of Theorem bnj79OLD

Step	Hyp	Ref	Expression
1		fneq2 4504	. . . . . 6 |- (x = w -> (z Fn x <-> z Fn w))
2	1	cbvabv 2420	. . . . 5 |- {x | z Fn x} = {w | z Fn w}
3	2	abeq1i 2002	. . . 4 |- (z Fn x <-> x e. {w | z Fn w})
4	3	sbbii 1538	. . 3 |- ([y / x]z Fn x <-> [y / x]x e. {w | z Fn w})
5		clelsb3 1990	. . 3 |- ([y / x]x e. {w | z Fn w} <-> y e. {w | z Fn w})
6		fneq2 4504	. . . . 5 |- (w = y -> (z Fn w <-> z Fn y))
7	6	cbvabv 2420	. . . 4 |- {w | z Fn w} = {y | z Fn y}
8	7	eleq2i 1961	. . 3 |- (y e. {w | z Fn w} <-> y e. {y | z Fn y})
9	4, 5, 8	3bitri 194	. 2 |- ([y / x]z Fn x <-> y e. {y | z Fn y})
10		abid 1873	. 2 |- (y e. {y | z Fn y} <-> z Fn y)
11	9, 10	bitri 190	1 |- ([y / x]z Fn x <-> z Fn y)

Syntax hints:   <-> wb 163   e. wcel 1300  [wsbc 1534  {cab 1871   Fn wfn 3993

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-fn 4009

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