Theorem isarep1 5672

Description: Part of a study of the Axiom of Replacement used by the Isabelle prover. The object PrimReplace is apparently the image of the function encoded by (x,) i.e. the class . If so, we can prove Isabelle's "Axiom of Replacement" conclusion without using the Axiom of Replacement, for which I (N. Megill) currently have no explanation. (Contributed by NM, 26-Oct-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)

Assertion

Ref	Expression
isarep1

Distinct variable groups:   ,   ,,

Proof of Theorem isarep1

Dummy variable is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3112	. . 3
2	1	elima 5347	. 2
3		df-br 4453	. . . 4
4		opelopabsb 4762	. . . 4
5		sbsbc 3331	. . . . . 6
6	5	sbbii 1746	. . . . 5
7		sbsbc 3331	. . . . 5
8	6, 7	bitr2i 250	. . . 4
9	3, 4, 8	3bitri 271	. . 3
10	9	rexbii 2959	. 2
11		nfs1v 2181	. . 3
12		nfv 1707	. . 3
13		sbequ12r 1993	. . 3
14	11, 12, 13	cbvrex 3081	. 2
15	2, 10, 14	3bitri 271	1

Syntax hints:  <->wb 184  [wsb 1739  e.wcel 1818  E.wrex 2808  [.wsbc 3327  <.cop 4035   class class class wbr 4452  {copab 4509  "cima 5007

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-br 4453  df-opab 4511  df-xp 5010  df-cnv 5012  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017