Theorem bnj90OLD 12446

Description: First-order logic and set theory.

Hypothesis

Ref	Expression
bnj90.1	|- Y e. _V

Assertion

Ref	Expression
bnj90OLD	|- ([Y / x]z Fn x <-> z Fn Y)

Distinct variable group:   x,z

Proof of Theorem bnj90OLD

Step	Hyp	Ref	Expression
1		bnj90.1	. . . 4 |- Y e. _V
2	1	isseti 2297	. . 3 |- E.y y = Y
3		bnj79 12440	. . . . 5 |- ([y / x]z Fn x <-> z Fn y)
4		dfsbcq 2455	. . . . . 6 |- (y = Y -> ([y / x]z Fn x <-> [Y / x]z Fn x))
5		fneq2 4504	. . . . . 6 |- (y = Y -> (z Fn y <-> z Fn Y))
6	4, 5	bibi12d 691	. . . . 5 |- (y = Y -> (([y / x]z Fn x <-> z Fn y) <-> ([Y / x]z Fn x <-> z Fn Y)))
7	3, 6	mpbii 210	. . . 4 |- (y = Y -> ([Y / x]z Fn x <-> z Fn Y))
8	7	eximi 1387	. . 3 |- (E.y y = Y -> E.y([Y / x]z Fn x <-> z Fn Y))
9	2, 8	ax-mp 7	. 2 |- E.y([Y / x]z Fn x <-> z Fn Y)
10		ax-17 1317	. . 3 |- (([Y / x]z Fn x <-> z Fn Y) -> A.y([Y / x]z Fn x <-> z Fn Y))
11	10	19.9 1383	. 2 |- (E.y([Y / x]z Fn x <-> z Fn Y) <-> ([Y / x]z Fn x <-> z Fn Y))
12	9, 11	mpbi 206	1 |- ([Y / x]z Fn x <-> z Fn Y)

Syntax hints:   <-> wb 163   = wceq 1298   e. wcel 1300  E.wex 1326  [wsbc 1534  _Vcvv 2292   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-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-v 2294  df-sbc 2454  df-fn 4009

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