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Theorem clos1eq2 5875
 Description: Equality law for closure. (Contributed by SF, 11-Feb-2015.)
Assertion
Ref Expression
clos1eq2 (R = T Clos1 (S, R) = Clos1 (S, T))

Proof of Theorem clos1eq2
Dummy variable a is distinct from all other variables.
StepHypRef Expression
1 imaeq1 4937 . . . . . 6 (R = T → (Ra) = (Ta))
21sseq1d 3298 . . . . 5 (R = T → ((Ra) a ↔ (Ta) a))
32anbi2d 684 . . . 4 (R = T → ((S a (Ra) a) ↔ (S a (Ta) a)))
43abbidv 2467 . . 3 (R = T → {a (S a (Ra) a)} = {a (S a (Ta) a)})
5 inteq 3929 . . 3 ({a (S a (Ra) a)} = {a (S a (Ta) a)} → {a (S a (Ra) a)} = {a (S a (Ta) a)})
64, 5syl 15 . 2 (R = T{a (S a (Ra) a)} = {a (S a (Ta) a)})
7 df-clos1 5873 . 2 Clos1 (S, R) = {a (S a (Ra) a)}
8 df-clos1 5873 . 2 Clos1 (S, T) = {a (S a (Ta) a)}
96, 7, 83eqtr4g 2410 1 (R = T Clos1 (S, R) = Clos1 (S, T))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 358   = wceq 1642  {cab 2339   ⊆ wss 3257  ∩cint 3926   “ cima 4722   Clos1 cclos1 5872 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-ral 2619  df-rex 2620  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-int 3927  df-br 4640  df-ima 4727  df-clos1 5873 This theorem is referenced by:  clos1exg  5877  clos1basesucg  5884  freceq12  6311
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