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Theorem ichid 46604
Description: A setvar variable is always interchangeable with itself. (Contributed by AV, 29-Jul-2023.)
Assertion
Ref Expression
ichid [𝑥𝑥]𝜑

Proof of Theorem ichid
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sbid 2239 . . . . 5 ([𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑎 / 𝑥]𝜑)
21sbbii 2071 . . . 4 ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑 ↔ [𝑥 / 𝑎][𝑎 / 𝑥]𝜑)
3 sbid2vw 2242 . . . 4 ([𝑥 / 𝑎][𝑎 / 𝑥]𝜑𝜑)
42, 3bitri 275 . . 3 ([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑)
54gen2 1790 . 2 𝑥𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑)
6 df-ich 46599 . 2 ([𝑥𝑥]𝜑 ↔ ∀𝑥𝑥([𝑥 / 𝑎][𝑥 / 𝑥][𝑎 / 𝑥]𝜑𝜑))
75, 6mpbir 230 1 [𝑥𝑥]𝜑
Colors of variables: wff setvar class
Syntax hints:  wb 205  wal 1531  [wsb 2059  [wich 46598
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-12 2163
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1774  df-sb 2060  df-ich 46599
This theorem is referenced by:  icheqid  46614
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