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Theorem ichv 43437
Description: Setvar variables are interchangeable in a wff they do not appear in. (Contributed by SN, 23-Nov-2023.)
Assertion
Ref Expression
ichv [𝑥𝑦]𝜑
Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Proof of Theorem ichv
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 sbv 2091 . . . . . . 7 ([𝑎 / 𝑦]𝜑𝜑)
21sbbii 2074 . . . . . 6 ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑦 / 𝑥]𝜑)
3 sbv 2091 . . . . . 6 ([𝑦 / 𝑥]𝜑𝜑)
42, 3bitri 276 . . . . 5 ([𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
54sbbii 2074 . . . 4 ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑 ↔ [𝑥 / 𝑎]𝜑)
6 sbv 2091 . . . 4 ([𝑥 / 𝑎]𝜑𝜑)
75, 6bitri 276 . . 3 ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
87gen2 1790 . 2 𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑)
9 df-ich 43434 . 2 ([𝑥𝑦]𝜑 ↔ ∀𝑥𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜑𝜑))
108, 9mpbir 232 1 [𝑥𝑦]𝜑
Colors of variables: wff setvar class
Syntax hints:  wb 207  wal 1528  [wsb 2062  [wich 43433
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 1904  ax-6 1963
This theorem depends on definitions:  df-bi 208  df-ex 1774  df-sb 2063  df-ich 43434
This theorem is referenced by: (None)
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