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Theorem sbcrel 5239
Description: Distribute proper substitution through a relation predicate. (Contributed by Alexander van der Vekens, 23-Jul-2017.)
Assertion
Ref Expression
sbcrel (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))

Proof of Theorem sbcrel
StepHypRef Expression
1 sbcssg 4118 . . 3 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V)))
2 csbconstg 3579 . . . 4 (𝐴𝑉𝐴 / 𝑥(V × V) = (V × V))
32sseq2d 3666 . . 3 (𝐴𝑉 → (𝐴 / 𝑥𝑅𝐴 / 𝑥(V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
41, 3bitrd 268 . 2 (𝐴𝑉 → ([𝐴 / 𝑥]𝑅 ⊆ (V × V) ↔ 𝐴 / 𝑥𝑅 ⊆ (V × V)))
5 df-rel 5150 . . 3 (Rel 𝑅𝑅 ⊆ (V × V))
65sbcbii 3524 . 2 ([𝐴 / 𝑥]Rel 𝑅[𝐴 / 𝑥]𝑅 ⊆ (V × V))
7 df-rel 5150 . 2 (Rel 𝐴 / 𝑥𝑅𝐴 / 𝑥𝑅 ⊆ (V × V))
84, 6, 73bitr4g 303 1 (𝐴𝑉 → ([𝐴 / 𝑥]Rel 𝑅 ↔ Rel 𝐴 / 𝑥𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wcel 2030  Vcvv 3231  [wsbc 3468  csb 3566  wss 3607   × cxp 5141  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-in 3614  df-ss 3621  df-nul 3949  df-rel 5150
This theorem is referenced by:  sbcfung  5950
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