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Theorem pw1in 4164
 Description: Unit power class distributes over intersection. (Contributed by SF, 13-Feb-2015.)
Assertion
Ref Expression
pw1in 1(AB) = (1A1B)

Proof of Theorem pw1in
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ancom 437 . . . . 5 (((y A x 1B) x = {y}) ↔ (x = {y} (y A x 1B)))
2 eleq1 2413 . . . . . . . . 9 (x = {y} → (x 1B ↔ {y} 1B))
3 snelpw1 4146 . . . . . . . . 9 ({y} 1By B)
42, 3syl6bb 252 . . . . . . . 8 (x = {y} → (x 1By B))
54anbi2d 684 . . . . . . 7 (x = {y} → ((y A x 1B) ↔ (y A y B)))
6 elin 3219 . . . . . . 7 (y (AB) ↔ (y A y B))
75, 6syl6bbr 254 . . . . . 6 (x = {y} → ((y A x 1B) ↔ y (AB)))
87pm5.32ri 619 . . . . 5 (((y A x 1B) x = {y}) ↔ (y (AB) x = {y}))
9 an12 772 . . . . 5 ((x = {y} (y A x 1B)) ↔ (y A (x = {y} x 1B)))
101, 8, 93bitr3i 266 . . . 4 ((y (AB) x = {y}) ↔ (y A (x = {y} x 1B)))
1110rexbii2 2643 . . 3 (y (AB)x = {y} ↔ y A (x = {y} x 1B))
12 elpw1 4144 . . 3 (x 1(AB) ↔ y (AB)x = {y})
13 elpw1 4144 . . . . 5 (x 1Ay A x = {y})
1413anbi1i 676 . . . 4 ((x 1A x 1B) ↔ (y A x = {y} x 1B))
15 elin 3219 . . . 4 (x (1A1B) ↔ (x 1A x 1B))
16 r19.41v 2764 . . . 4 (y A (x = {y} x 1B) ↔ (y A x = {y} x 1B))
1714, 15, 163bitr4i 268 . . 3 (x (1A1B) ↔ y A (x = {y} x 1B))
1811, 12, 173bitr4i 268 . 2 (x 1(AB) ↔ x (1A1B))
1918eqriv 2350 1 1(AB) = (1A1B)
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∃wrex 2615   ∩ cin 3208  {csn 3737  ℘1cpw1 4135 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  ax-nin 4078  ax-sn 4087 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-ne 2518  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-pw 3724  df-sn 3741  df-1c 4136  df-pw1 4137 This theorem is referenced by:  tfindi  4496  tcdi  6164  ce0addcnnul  6179
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