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Theorem ichan 46123
Description: If two setvar variables are interchangeable in two wffs, then they are interchangeable in the conjunction of these two wffs. Notice that the reverse implication is not necessarily true. Corresponding theorems will hold for other commutative operations, too. (Contributed by AV, 31-Jul-2023.) Use df-ich 46114 instead of dfich2 46126 to reduce axioms. (Revised by SN, 4-May-2024.)
Assertion
Ref Expression
ichan (([𝑎𝑏]𝜑 ∧ [𝑎𝑏]𝜓) → [𝑎𝑏](𝜑𝜓))

Proof of Theorem ichan
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sban 2084 . . . . . . . 8 ([𝑥 / 𝑏](𝜑𝜓) ↔ ([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
21sbbii 2080 . . . . . . 7 ([𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ [𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
32sbbii 2080 . . . . . 6 ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
4 sban 2084 . . . . . . 7 ([𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ ([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
54sbbii 2080 . . . . . 6 ([𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ [𝑎 / 𝑥]([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
6 sban 2084 . . . . . 6 ([𝑎 / 𝑥]([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
73, 5, 63bitri 297 . . . . 5 ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
8 pm4.38 637 . . . . 5 ((([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑) ∧ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓)) → (([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓) ↔ (𝜑𝜓)))
97, 8bitrid 283 . . . 4 ((([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑) ∧ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓)) → ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ (𝜑𝜓)))
109alanimi 1819 . . 3 ((∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑) ∧ ∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓)) → ∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ (𝜑𝜓)))
1110alanimi 1819 . 2 ((∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑) ∧ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓)) → ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ (𝜑𝜓)))
12 df-ich 46114 . . 3 ([𝑎𝑏]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑))
13 df-ich 46114 . . 3 ([𝑎𝑏]𝜓 ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓))
1412, 13anbi12i 628 . 2 (([𝑎𝑏]𝜑 ∧ [𝑎𝑏]𝜓) ↔ (∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑𝜑) ∧ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓𝜓)))
15 df-ich 46114 . 2 ([𝑎𝑏](𝜑𝜓) ↔ ∀𝑎𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑𝜓) ↔ (𝜑𝜓)))
1611, 14, 153imtr4i 292 1 (([𝑎𝑏]𝜑 ∧ [𝑎𝑏]𝜓) → [𝑎𝑏](𝜑𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wal 1540  [wsb 2068  [wich 46113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812
This theorem depends on definitions:  df-bi 206  df-an 398  df-sb 2069  df-ich 46114
This theorem is referenced by:  ichim  46125
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