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Theorem intab 3635
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 2043 . . . . . . . . . 10 (z = x → (z = Ax = A))
21anbi2d 437 . . . . . . . . 9 (z = x → ((φ z = A) ↔ (φ x = A)))
32exbidv 1703 . . . . . . . 8 (z = x → (y(φ z = A) ↔ y(φ x = A)))
43cbvabv 2158 . . . . . . 7 {zy(φ z = A)} = {xy(φ x = A)}
5 intab.2 . . . . . . 7 {xy(φ x = A)} V
64, 5eqeltri 2107 . . . . . 6 {zy(φ z = A)} V
7 nfe1 1382 . . . . . . . . 9 yy(φ z = A)
87nfab 2179 . . . . . . . 8 y{zy(φ z = A)}
98nfeq2 2186 . . . . . . 7 y x = {zy(φ z = A)}
10 eleq2 2098 . . . . . . . 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 1503 . . . . . 6 (x = {zy(φ z = A)} → (y(φA x) ↔ y(φA {zy(φ z = A)})))
136, 12elab 2681 . . . . 5 ({zy(φ z = A)} {xy(φA x)} ↔ y(φA {zy(φ z = A)}))
14 19.8a 1479 . . . . . . . . 9 ((φ z = A) → y(φ z = A))
1514ex 108 . . . . . . . 8 (φ → (z = Ay(φ z = A)))
1615alrimiv 1751 . . . . . . 7 (φz(z = Ay(φ z = A)))
17 intab.1 . . . . . . . 8 A V
1817sbc6 2783 . . . . . . 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 2759 . . . . . 6 ([A / z]y(φ z = A) ↔ A {zy(φ z = A)})
2119, 20sylib 127 . . . . 5 (φA {zy(φ z = A)})
2213, 21mpgbir 1339 . . . 4 {zy(φ z = A)} {xy(φA x)}
23 intss1 3621 . . . 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 1509 . . . . . . . 8 ((y(φ z = A) y(φA x)) → y((φ z = A) (φA x)))
26 simplr 482 . . . . . . . . . 10 (((φ z = A) (φA x)) → z = A)
27 pm3.35 329 . . . . . . . . . . 11 ((φ (φA x)) → A x)
2827adantlr 446 . . . . . . . . . 10 (((φ z = A) (φA x)) → A x)
2926, 28eqeltrd 2111 . . . . . . . . 9 (((φ z = A) (φA x)) → z x)
3029exlimiv 1486 . . . . . . . 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 1751 . . . . 5 (y(φ z = A) → x(y(φA x) → z x))
34 vex 2554 . . . . . 6 z V
3534elintab 3617 . . . . 5 (z {xy(φA x)} ↔ x(y(φA x) → z x))
3633, 35sylibr 137 . . . 4 (y(φ z = A) → z {xy(φA x)})
3736abssi 3009 . . 3 {zy(φ z = A)} ⊆ {xy(φA x)}
3824, 37eqssi 2955 . 2 {xy(φA x)} = {zy(φ z = A)}
3938, 4eqtri 2057 1 {xy(φA x)} = {xy(φ x = A)}
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wal 1240   = wceq 1242  wex 1378   wcel 1390  {cab 2023  Vcvv 2551  [wsbc 2758  wss 2911   cint 3606
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-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-in 2918  df-ss 2925  df-int 3607
This theorem is referenced by: (None)
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