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Theorem intab 3956
 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). Typically, abrexex2 in set.mm or abexssex in set.mm can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Hypotheses
Ref Expression
intab.1 A V
intab.2 {x y(φ x = A)} V
Assertion
Ref Expression
intab {x y(φA x)} = {x y(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 2359 . . . . . . . . . 10 (z = x → (z = Ax = A))
21anbi2d 684 . . . . . . . . 9 (z = x → ((φ z = A) ↔ (φ x = A)))
32exbidv 1626 . . . . . . . 8 (z = x → (y(φ z = A) ↔ y(φ x = A)))
43cbvabv 2472 . . . . . . 7 {z y(φ z = A)} = {x y(φ x = A)}
5 intab.2 . . . . . . 7 {x y(φ x = A)} V
64, 5eqeltri 2423 . . . . . 6 {z y(φ z = A)} V
7 nfe1 1732 . . . . . . . . 9 yy(φ z = A)
87nfab 2493 . . . . . . . 8 y{z y(φ z = A)}
98nfeq2 2500 . . . . . . 7 y x = {z y(φ z = A)}
10 eleq2 2414 . . . . . . . 8 (x = {z y(φ z = A)} → (A xA {z y(φ z = A)}))
1110imbi2d 307 . . . . . . 7 (x = {z y(φ z = A)} → ((φA x) ↔ (φA {z y(φ z = A)})))
129, 11albid 1772 . . . . . 6 (x = {z y(φ z = A)} → (y(φA x) ↔ y(φA {z y(φ z = A)})))
136, 12elab 2985 . . . . 5 ({z y(φ z = A)} {x y(φA x)} ↔ y(φA {z y(φ z = A)}))
14 19.8a 1756 . . . . . . . . 9 ((φ z = A) → y(φ z = A))
1514ex 423 . . . . . . . 8 (φ → (z = Ay(φ z = A)))
1615alrimiv 1631 . . . . . . 7 (φz(z = Ay(φ z = A)))
17 intab.1 . . . . . . . 8 A V
1817sbc6 3072 . . . . . . 7 ([̣A / zy(φ z = A) ↔ z(z = Ay(φ z = A)))
1916, 18sylibr 203 . . . . . 6 (φ → [̣A / zy(φ z = A))
20 df-sbc 3047 . . . . . 6 ([̣A / zy(φ z = A) ↔ A {z y(φ z = A)})
2119, 20sylib 188 . . . . 5 (φA {z y(φ z = A)})
2213, 21mpgbir 1550 . . . 4 {z y(φ z = A)} {x y(φA x)}
23 intss1 3941 . . . 4 ({z y(φ z = A)} {x y(φA x)} → {x y(φA x)} {z y(φ z = A)})
2422, 23ax-mp 8 . . 3 {x y(φA x)} {z y(φ z = A)}
25 19.29r 1597 . . . . . . . 8 ((y(φ z = A) y(φA x)) → y((φ z = A) (φA x)))
26 simplr 731 . . . . . . . . . 10 (((φ z = A) (φA x)) → z = A)
27 pm3.35 570 . . . . . . . . . . 11 ((φ (φA x)) → A x)
2827adantlr 695 . . . . . . . . . 10 (((φ z = A) (φA x)) → A x)
2926, 28eqeltrd 2427 . . . . . . . . 9 (((φ z = A) (φA x)) → z x)
3029exlimiv 1634 . . . . . . . 8 (y((φ z = A) (φA x)) → z x)
3125, 30syl 15 . . . . . . 7 ((y(φ z = A) y(φA x)) → z x)
3231ex 423 . . . . . 6 (y(φ z = A) → (y(φA x) → z x))
3332alrimiv 1631 . . . . 5 (y(φ z = A) → x(y(φA x) → z x))
34 vex 2862 . . . . . 6 z V
3534elintab 3937 . . . . 5 (z {x y(φA x)} ↔ x(y(φA x) → z x))
3633, 35sylibr 203 . . . 4 (y(φ z = A) → z {x y(φA x)})
3736abssi 3341 . . 3 {z y(φ z = A)} {x y(φA x)}
3824, 37eqssi 3288 . 2 {x y(φA x)} = {z y(φ z = A)}
3938, 4eqtri 2373 1 {x y(φA x)} = {x y(φ x = A)}
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358  ∀wal 1540  ∃wex 1541   = wceq 1642   ∈ wcel 1710  {cab 2339  Vcvv 2859  [̣wsbc 3046   ⊆ wss 3257  ∩cint 3926 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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-nan 1288  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-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927 This theorem is referenced by: (None)
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