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Theorem cbvrab 2549
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 Aφ} = {y Aψ}

Proof of Theorem cbvrab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 nfv 1418 . . . 4 z(x A φ)
2 cbvrab.1 . . . . . 6 xA
32nfcri 2169 . . . . 5 x z A
4 nfs1v 1812 . . . . 5 x[z / x]φ
53, 4nfan 1454 . . . 4 x(z A [z / x]φ)
6 eleq1 2097 . . . . 5 (x = z → (x Az A))
7 sbequ12 1651 . . . . 5 (x = z → (φ ↔ [z / x]φ))
86, 7anbi12d 442 . . . 4 (x = z → ((x A φ) ↔ (z A [z / x]φ)))
91, 5, 8cbvab 2157 . . 3 {x ∣ (x A φ)} = {z ∣ (z A [z / x]φ)}
10 cbvrab.2 . . . . . 6 yA
1110nfcri 2169 . . . . 5 y z A
12 cbvrab.3 . . . . . 6 yφ
1312nfsb 1819 . . . . 5 y[z / x]φ
1411, 13nfan 1454 . . . 4 y(z A [z / x]φ)
15 nfv 1418 . . . 4 z(y A ψ)
16 eleq1 2097 . . . . 5 (z = y → (z Ay A))
17 sbequ 1718 . . . . . 6 (z = y → ([z / x]φ ↔ [y / x]φ))
18 cbvrab.4 . . . . . . 7 xψ
19 cbvrab.5 . . . . . . 7 (x = y → (φψ))
2018, 19sbie 1671 . . . . . 6 ([y / x]φψ)
2117, 20syl6bb 185 . . . . 5 (z = y → ([z / x]φψ))
2216, 21anbi12d 442 . . . 4 (z = y → ((z A [z / x]φ) ↔ (y A ψ)))
2314, 15, 22cbvab 2157 . . 3 {z ∣ (z A [z / x]φ)} = {y ∣ (y A ψ)}
249, 23eqtri 2057 . 2 {x ∣ (x A φ)} = {y ∣ (y A ψ)}
25 df-rab 2309 . 2 {x Aφ} = {x ∣ (x A φ)}
26 df-rab 2309 . 2 {y Aψ} = {y ∣ (y A ψ)}
2724, 25, 263eqtr4i 2067 1 {x Aφ} = {y Aψ}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98   = wceq 1242  wnf 1346   wcel 1390  [wsb 1642  {cab 2023  wnfc 2162  {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309
This theorem is referenced by:  cbvrabv  2550  elrabsf  2795  tfis  4249
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