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Theorem csbtt 3750
Description: Substitution doesn't affect a constant 𝐵 (in which 𝑥 is not free). (Contributed by Mario Carneiro, 14-Oct-2016.)
Assertion
Ref Expression
csbtt ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)

Proof of Theorem csbtt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 df-csb 3740 . 2 𝐴 / 𝑥𝐵 = {𝑦[𝐴 / 𝑥]𝑦𝐵}
2 nfcr 2951 . . . 4 (𝑥𝐵 → Ⅎ𝑥 𝑦𝐵)
3 sbctt 3707 . . . 4 ((𝐴𝑉 ∧ Ⅎ𝑥 𝑦𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
42, 3sylan2 582 . . 3 ((𝐴𝑉𝑥𝐵) → ([𝐴 / 𝑥]𝑦𝐵𝑦𝐵))
54abbi1dv 2938 . 2 ((𝐴𝑉𝑥𝐵) → {𝑦[𝐴 / 𝑥]𝑦𝐵} = 𝐵)
61, 5syl5eq 2863 1 ((𝐴𝑉𝑥𝐵) → 𝐴 / 𝑥𝐵 = 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1637  wnf 1863  wcel 2157  {cab 2803  wnfc 2946  [wsbc 3644  csb 3739
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-v 3404  df-sbc 3645  df-csb 3740
This theorem is referenced by:  csbconstgf  3751  sbnfc2  4216  constlimc  40354
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