New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  cbvrab GIF version

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 xA
cbvrab.2 yA
cbvrab.3 yφ
cbvrab.4 xψ
cbvrab.5 (x = y → (φψ))
Assertion
Ref Expression
cbvrab {x A φ} = {y A ψ}

Proof of Theorem cbvrab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1619 . . . 4 z(x A φ)
2 cbvrab.1 . . . . . 6 xA
32nfcri 2483 . . . . 5 x z A
4 nfs1v 2106 . . . . 5 x[z / x]φ
53, 4nfan 1824 . . . 4 x(z A [z / x]φ)
6 eleq1 2413 . . . . 5 (x = z → (x Az A))
7 sbequ12 1919 . . . . 5 (x = z → (φ ↔ [z / x]φ))
86, 7anbi12d 691 . . . 4 (x = z → ((x A φ) ↔ (z A [z / x]φ)))
91, 5, 8cbvab 2471 . . 3 {x (x A φ)} = {z (z A [z / x]φ)}
10 cbvrab.2 . . . . . 6 yA
1110nfcri 2483 . . . . 5 y z A
12 cbvrab.3 . . . . . 6 yφ
1312nfsb 2109 . . . . 5 y[z / x]φ
1411, 13nfan 1824 . . . 4 y(z A [z / x]φ)
15 nfv 1619 . . . 4 z(y A ψ)
16 eleq1 2413 . . . . 5 (z = y → (z Ay A))
17 sbequ 2060 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
18 cbvrab.4 . . . . . . 7 xψ
19 cbvrab.5 . . . . . . 7 (x = y → (φψ))
2018, 19sbie 2038 . . . . . 6 ([y / x]φψ)
2117, 20syl6bb 252 . . . . 5 (z = y → ([z / x]φψ))
2216, 21anbi12d 691 . . . 4 (z = y → ((z A [z / x]φ) ↔ (y A ψ)))
2314, 15, 22cbvab 2471 . . 3 {z (z A [z / x]φ)} = {y (y A ψ)}
249, 23eqtri 2373 . 2 {x (x A φ)} = {y (y A ψ)}
25 df-rab 2623 . 2 {x A φ} = {x (x A φ)}
26 df-rab 2623 . 2 {y A ψ} = {y (y A ψ)}
2724, 25, 263eqtr4i 2383 1 {x A φ} = {y A ψ}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  Ⅎwnf 1544   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  Ⅎwnfc 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
 Copyright terms: Public domain W3C validator