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Theorem intab 3618
Description: The intersection of a special case of a class abstraction. y may be free in φ and A, which can be thought of a φ(y) and A(y). (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1 A V
intab.2 {xy(φ x = A)} V
Assertion
Ref Expression
intab {xy(φA x)} = {xy(φ x = A)}
Distinct variable groups:   x,A   φ,x   x,y
Allowed substitution hints:   φ(y)   A(y)

Proof of Theorem intab
Dummy variable z is distinct from all other variables.
StepHypRef Expression
1 eqeq1 2028 . . . . . . . . . 10 (z = x → (z = Ax = A))
21anbi2d 440 . . . . . . . . 9 (z = x → ((φ z = A) ↔ (φ x = A)))
32exbidv 1688 . . . . . . . 8 (z = x → (y(φ z = A) ↔ y(φ x = A)))
43cbvabv 2143 . . . . . . 7 {zy(φ z = A)} = {xy(φ x = A)}
5 intab.2 . . . . . . 7 {xy(φ x = A)} V
64, 5eqeltri 2092 . . . . . 6 {zy(φ z = A)} V
7 nfe1 1366 . . . . . . . . 9 yy(φ z = A)
87nfab 2164 . . . . . . . 8 y{zy(φ z = A)}
98nfeq2 2171 . . . . . . 7 y x = {zy(φ z = A)}
10 eleq2 2083 . . . . . . . 8 (x = {zy(φ z = A)} → (A xA {zy(φ z = A)}))
1110imbi2d 219 . . . . . . 7 (x = {zy(φ z = A)} → ((φA x) ↔ (φA {zy(φ z = A)})))
129, 11albid 1488 . . . . . 6 (x = {zy(φ z = A)} → (y(φA x) ↔ y(φA {zy(φ z = A)})))
136, 12elab 2664 . . . . 5 ({zy(φ z = A)} {xy(φA x)} ↔ y(φA {zy(φ z = A)}))
14 19.8a 1464 . . . . . . . . 9 ((φ z = A) → y(φ z = A))
1514ex 108 . . . . . . . 8 (φ → (z = Ay(φ z = A)))
1615alrimiv 1736 . . . . . . 7 (φz(z = Ay(φ z = A)))
17 intab.1 . . . . . . . 8 A V
1817sbc6 2766 . . . . . . 7 ([A / z]y(φ z = A) ↔ z(z = Ay(φ z = A)))
1916, 18sylibr 137 . . . . . 6 (φ[A / z]y(φ z = A))
20 df-sbc 2742 . . . . . 6 ([A / z]y(φ z = A) ↔ A {zy(φ z = A)})
2119, 20sylib 127 . . . . 5 (φA {zy(φ z = A)})
2213, 21mpgbir 1322 . . . 4 {zy(φ z = A)} {xy(φA x)}
23 intss1 3604 . . . 4 ({zy(φ z = A)} {xy(φA x)} → {xy(φA x)} ⊆ {zy(φ z = A)})
2422, 23ax-mp 7 . . 3 {xy(φA x)} ⊆ {zy(φ z = A)}
25 19.29r 1494 . . . . . . . 8 ((y(φ z = A) y(φA x)) → y((φ z = A) (φA x)))
26 simplr 470 . . . . . . . . . 10 (((φ z = A) (φA x)) → z = A)
27 pm3.35 329 . . . . . . . . . . 11 ((φ (φA x)) → A x)
2827adantlr 449 . . . . . . . . . 10 (((φ z = A) (φA x)) → A x)
2926, 28eqeltrd 2096 . . . . . . . . 9 (((φ z = A) (φA x)) → z x)
3029exlimiv 1471 . . . . . . . 8 (y((φ z = A) (φA x)) → z x)
3125, 30syl 14 . . . . . . 7 ((y(φ z = A) y(φA x)) → z x)
3231ex 108 . . . . . 6 (y(φ z = A) → (y(φA x) → z x))
3332alrimiv 1736 . . . . 5 (y(φ z = A) → x(y(φA x) → z x))
34 vex 2538 . . . . . 6 z V
3534elintab 3600 . . . . 5 (z {xy(φA x)} ↔ x(y(φA x) → z x))
3633, 35sylibr 137 . . . 4 (y(φ z = A) → z {xy(φA x)})
3736abssi 2992 . . 3 {zy(φ z = A)} ⊆ {xy(φA x)}
3824, 37eqssi 2938 . 2 {xy(φA x)} = {zy(φ z = A)}
3938, 4eqtri 2042 1 {xy(φA x)} = {xy(φ x = A)}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1226   = wceq 1228  wex 1362   wcel 1374  {cab 2008  Vcvv 2535  [wsbc 2741  wss 2894   cint 3589
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-sbc 2742  df-in 2901  df-ss 2908  df-int 3590
This theorem is referenced by: (None)
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