Theorem cbvrab 2857

Description: Rule to change the bound variable in a restricted class abstraction, using implicit substitution. This version has bound-variable hypotheses in place of distinct variable conditions. (Contributed by Andrew Salmon, 11-Jul-2011.) (Revised by Mario Carneiro, 9-Oct-2016.)

Hypotheses

Ref	Expression
cbvrab.1	|- F/_xA
cbvrab.2	|- F/_yA
cbvrab.3	|- F/yph
cbvrab.4	|- F/xps
cbvrab.5	|- (x = y -> (ph <-> ps))

Assertion

Ref	Expression
cbvrab	|- {x e. A | ph} = {y e. A | ps}

Proof of Theorem cbvrab

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 |- F/z(x e. A /\ ph)
2		cbvrab.1	. . . . . 6 |- F/_xA
3	2	nfcri 2483	. . . . 5 |- F/x z e. A
4		nfs1v 2106	. . . . 5 |- F/x[z / x]ph
5	3, 4	nfan 1824	. . . 4 |- F/x(z e. A /\ [z / x]ph)
6		eleq1 2413	. . . . 5 |- (x = z -> (x e. A <-> z e. A))
7		sbequ12 1919	. . . . 5 |- (x = z -> (ph <-> [z / x]ph))
8	6, 7	anbi12d 691	. . . 4 |- (x = z -> ((x e. A /\ ph) <-> (z e. A /\ [z / x]ph)))
9	1, 5, 8	cbvab 2471	. . 3 |- {x | (x e. A /\ ph)} = {z | (z e. A /\ [z / x]ph)}
10		cbvrab.2	. . . . . 6 |- F/_yA
11	10	nfcri 2483	. . . . 5 |- F/y z e. A
12		cbvrab.3	. . . . . 6 |- F/yph
13	12	nfsb 2109	. . . . 5 |- F/y[z / x]ph
14	11, 13	nfan 1824	. . . 4 |- F/y(z e. A /\ [z / x]ph)
15		nfv 1619	. . . 4 |- F/z(y e. A /\ ps)
16		eleq1 2413	. . . . 5 |- (z = y -> (z e. A <-> y e. A))
17		sbequ 2060	. . . . . 6 |- (z = y -> ([z / x]ph <-> [y / x]ph))
18		cbvrab.4	. . . . . . 7 |- F/xps
19		cbvrab.5	. . . . . . 7 |- (x = y -> (ph <-> ps))
20	18, 19	sbie 2038	. . . . . 6 |- ([y / x]ph <-> ps)
21	17, 20	syl6bb 252	. . . . 5 |- (z = y -> ([z / x]ph <-> ps))
22	16, 21	anbi12d 691	. . . 4 |- (z = y -> ((z e. A /\ [z / x]ph) <-> (y e. A /\ ps)))
23	14, 15, 22	cbvab 2471	. . 3 |- {z | (z e. A /\ [z / x]ph)} = {y | (y e. A /\ ps)}
24	9, 23	eqtri 2373	. 2 |- {x | (x e. A /\ ph)} = {y | (y e. A /\ ps)}
25		df-rab 2623	. 2 |- {x e. A | ph} = {x | (x e. A /\ ph)}
26		df-rab 2623	. 2 |- {y e. A | ps} = {y | (y e. A /\ ps)}
27	24, 25, 26	3eqtr4i 2383	1 |- {x e. A | ph} = {y e. A | ps}

Syntax hints:   -> wi 4   <-> wb 176   /\ wa 358   F/wnf 1544   = wceq 1642  [wsb 1648   e. wcel 1710  {cab 2339   F/_wnfc 2476  {crab 2618

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-rab 2623

This theorem is referenced by:  cbvrabv  2858  elrabsf  3084