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Theorem ichal 43998
 Description: Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)
Assertion
Ref Expression
ichal (∀𝑥[𝑎𝑏]𝜑 → [𝑎𝑏]∀𝑥𝜑)
Distinct variable groups:   𝑥,𝑎   𝑥,𝑏
Allowed substitution hints:   𝜑(𝑥,𝑎,𝑏)

Proof of Theorem ichal
Dummy variable 𝑢 is distinct from all other variables.
StepHypRef Expression
1 ax-11 2158 . . 3 (∀𝑥𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ∀𝑎𝑥𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
2 ax-11 2158 . . . 4 (∀𝑥𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ∀𝑏𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
32alimi 1813 . . 3 (∀𝑎𝑥𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ∀𝑎𝑏𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
4 sbal 2163 . . . . . . 7 ([𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑏]𝜑)
542sbbii 2082 . . . . . 6 ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ [𝑎 / 𝑢][𝑏 / 𝑎]∀𝑥[𝑢 / 𝑏]𝜑)
6 sbal 2163 . . . . . . 7 ([𝑏 / 𝑎]∀𝑥[𝑢 / 𝑏]𝜑 ↔ ∀𝑥[𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
76sbbii 2081 . . . . . 6 ([𝑎 / 𝑢][𝑏 / 𝑎]∀𝑥[𝑢 / 𝑏]𝜑 ↔ [𝑎 / 𝑢]∀𝑥[𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
8 sbal 2163 . . . . . 6 ([𝑎 / 𝑢]∀𝑥[𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ∀𝑥[𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
95, 7, 83bitri 300 . . . . 5 ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥[𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
10 albi 1820 . . . . 5 (∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → (∀𝑥[𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ∀𝑥𝜑))
119, 10syl5bb 286 . . . 4 (∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
12112alimi 1814 . . 3 (∀𝑎𝑏𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
131, 3, 123syl 18 . 2 (∀𝑥𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑) → ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
14 df-ich 43978 . . 3 ([𝑎𝑏]𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
1514albii 1821 . 2 (∀𝑥[𝑎𝑏]𝜑 ↔ ∀𝑥𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑𝜑))
16 df-ich 43978 . 2 ([𝑎𝑏]∀𝑥𝜑 ↔ ∀𝑎𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
1713, 15, 163imtr4i 295 1 (∀𝑥[𝑎𝑏]𝜑 → [𝑎𝑏]∀𝑥𝜑)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069  [wich 43977 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-11 2158 This theorem depends on definitions:  df-bi 210  df-ex 1782  df-sb 2070  df-ich 43978 This theorem is referenced by: (None)
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