Theorem axrep5 3433

Description: Axiom of Replacement (similar to Axiom Rep of [BellMachover] p. 463). The antecedent tells us ph is analogous to a "function" from x to y (although it is not really a function since it is a wff and not a class). In the consequent we postulate the existence of a set z that corresponds to the "image" of ph restricted to some other set w. The hypothesis says z must not be free in ph.

Hypothesis

Ref	Expression
axrep5.1	|- (ph -> A.zph)

Assertion

Ref	Expression
axrep5	|- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Distinct variable group:   x,y,z,w

Proof of Theorem axrep5

Step	Hyp	Ref	Expression
1		19.37v 1683	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> (x e. w -> E.zA.y(ph -> y = z)))
2		impexp 374	. . . . . . . 8 |- (((x e. w /\ ph) -> y = z) <-> (x e. w -> (ph -> y = z)))
3	2	albii 1346	. . . . . . 7 |- (A.y((x e. w /\ ph) -> y = z) <-> A.y(x e. w -> (ph -> y = z)))
4		19.21v 1663	. . . . . . 7 |- (A.y(x e. w -> (ph -> y = z)) <-> (x e. w -> A.y(ph -> y = z)))
5	3, 4	bitr2i 191	. . . . . 6 |- ((x e. w -> A.y(ph -> y = z)) <-> A.y((x e. w /\ ph) -> y = z))
6	5	exbii 1398	. . . . 5 |- (E.z(x e. w -> A.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
7	1, 6	bitr3i 192	. . . 4 |- ((x e. w -> E.zA.y(ph -> y = z)) <-> E.zA.y((x e. w /\ ph) -> y = z))
8	7	albii 1346	. . 3 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) <-> A.xE.zA.y((x e. w /\ ph) -> y = z))
9		ax-17 1317	. . . . 5 |- (x e. w -> A.z x e. w)
10		axrep5.1	. . . . 5 |- (ph -> A.zph)
11	9, 10	hban 1356	. . . 4 |- ((x e. w /\ ph) -> A.z(x e. w /\ ph))
12	11	axrep4 3432	. . 3 |- (A.xE.zA.y((x e. w /\ ph) -> y = z) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
13	8, 12	sylbi 216	. 2 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))))
14		anabs5 551	. . . . . 6 |- ((x e. w /\ (x e. w /\ ph)) <-> (x e. w /\ ph))
15	14	exbii 1398	. . . . 5 |- (E.x(x e. w /\ (x e. w /\ ph)) <-> E.x(x e. w /\ ph))
16	15	bibi2i 669	. . . 4 |- ((y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> (y e. z <-> E.x(x e. w /\ ph)))
17	16	albii 1346	. . 3 |- (A.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> A.y(y e. z <-> E.x(x e. w /\ ph)))
18	17	exbii 1398	. 2 |- (E.zA.y(y e. z <-> E.x(x e. w /\ (x e. w /\ ph))) <-> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))
19	13, 18	sylib 215	1 |- (A.x(x e. w -> E.zA.y(ph -> y = z)) -> E.zA.y(y e. z <-> E.x(x e. w /\ ph)))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326

This theorem is referenced by:  zfrepclf 3434  axsep 3437

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-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-rep 3428

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327

Copyright terms: Public domain