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Theorem sbcel12g 3151
 Description: Distribute proper substitution through a membership relation. (Contributed by NM, 10-Nov-2005.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)
Assertion
Ref Expression
sbcel12g (A V → ([̣A / xB C[A / x]B [A / x]C))

Proof of Theorem sbcel12g
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfsbcq2 3049 . . 3 (z = A → ([z / x]B C ↔ [̣A / xB C))
2 dfsbcq2 3049 . . . . 5 (z = A → ([z / x]y B ↔ [̣A / xy B))
32abbidv 2467 . . . 4 (z = A → {y [z / x]y B} = {y A / xy B})
4 dfsbcq2 3049 . . . . 5 (z = A → ([z / x]y C ↔ [̣A / xy C))
54abbidv 2467 . . . 4 (z = A → {y [z / x]y C} = {y A / xy C})
63, 5eleq12d 2421 . . 3 (z = A → ({y [z / x]y B} {y [z / x]y C} ↔ {y A / xy B} {y A / xy C}))
7 nfs1v 2106 . . . . . 6 x[z / x]y B
87nfab 2493 . . . . 5 x{y [z / x]y B}
9 nfs1v 2106 . . . . . 6 x[z / x]y C
109nfab 2493 . . . . 5 x{y [z / x]y C}
118, 10nfel 2497 . . . 4 x{y [z / x]y B} {y [z / x]y C}
12 sbab 2475 . . . . 5 (x = zB = {y [z / x]y B})
13 sbab 2475 . . . . 5 (x = zC = {y [z / x]y C})
1412, 13eleq12d 2421 . . . 4 (x = z → (B C ↔ {y [z / x]y B} {y [z / x]y C}))
1511, 14sbie 2038 . . 3 ([z / x]B C ↔ {y [z / x]y B} {y [z / x]y C})
161, 6, 15vtoclbg 2915 . 2 (A V → ([̣A / xB C ↔ {y A / xy B} {y A / xy C}))
17 df-csb 3137 . . 3 [A / x]B = {y A / xy B}
18 df-csb 3137 . . 3 [A / x]C = {y A / xy C}
1917, 18eleq12i 2418 . 2 ([A / x]B [A / x]C ↔ {y A / xy B} {y A / xy C})
2016, 19syl6bbr 254 1 (A V → ([̣A / xB C[A / x]B [A / x]C))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339  [̣wsbc 3046  [csb 3136 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-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-csb 3137 This theorem is referenced by:  sbcnel12g  3153  sbcel1g  3155  sbcel2g  3157  sbccsb2g  3165
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