Theorem bnj78 12439

Description: First-order logic and set theory.

Assertion

Ref	Expression
bnj78	|- ([y / x]x Fn A <-> y Fn A)

Distinct variable group:   x,A,y

Proof of Theorem bnj78

Step	Hyp	Ref	Expression
1		clelsb3 1990	. . 3 |- ([y / x]x e. {z | z Fn A} <-> y e. {z | z Fn A})
2		fneq1 4503	. . . . . 6 |- (x = z -> (x Fn A <-> z Fn A))
3	2	cbvabv 2420	. . . . 5 |- {x | x Fn A} = {z | z Fn A}
4	3	abeq1i 2002	. . . 4 |- (x Fn A <-> x e. {z | z Fn A})
5	4	sbbii 1538	. . 3 |- ([y / x]x Fn A <-> [y / x]x e. {z | z Fn A})
6		fneq1 4503	. . . . . 6 |- (x = y -> (x Fn A <-> y Fn A))
7	6	cbvabv 2420	. . . . 5 |- {x | x Fn A} = {y | y Fn A}
8	7, 3	eqtr3i 1910	. . . 4 |- {y | y Fn A} = {z | z Fn A}
9	8	eleq2i 1961	. . 3 |- (y e. {y | y Fn A} <-> y e. {z | z Fn A})
10	1, 5, 9	3bitr4i 200	. 2 |- ([y / x]x Fn A <-> y e. {y | y Fn A})
11		abid 1873	. 2 |- (y e. {y | y Fn A} <-> y Fn A)
12	10, 11	bitri 190	1 |- ([y / x]x Fn A <-> y Fn A)

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-14 1312  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  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-br 3339  df-opab 3396  df-id 3586  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-fun 4008  df-fn 4009

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