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Theorem sbss 4515
Description: Set substitution into the first argument of a subset relation. (Contributed by Rodolfo Medina, 7-Jul-2010.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)
Assertion
Ref Expression
sbss ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem sbss
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 sseq1 4000 . 2 (𝑥 = 𝑧 → (𝑥𝐴𝑧𝐴))
2 sseq1 4000 . 2 (𝑧 = 𝑦 → (𝑧𝐴𝑦𝐴))
31, 2sbievw2 2091 1 ([𝑦 / 𝑥]𝑥𝐴𝑦𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  [wsb 2059  wss 3941
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-in 3948  df-ss 3958
This theorem is referenced by: (None)
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