Theorem ichal 44591

Description: Move a universal quantifier inside interchangeability. (Contributed by SN, 30-Aug-2023.)

Assertion

Ref	Expression
ichal	⊢ (∀𝑥[𝑎⇄𝑏]𝜑 → [𝑎⇄𝑏]∀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.

Step	Hyp	Ref	Expression
1		ax-11 2158	. . 3 ⊢ (∀𝑥∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ∀𝑎∀𝑥∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
2		ax-11 2158	. . . 4 ⊢ (∀𝑥∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ∀𝑏∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
3	2	alimi 1819	. . 3 ⊢ (∀𝑎∀𝑥∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ∀𝑎∀𝑏∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
4		sbal 2163	. . . . . . 7 ⊢ ([𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥[𝑢 / 𝑏]𝜑)
5	4	2sbbii 2083	. . . . . 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)
7	6	sbbii 2082	. . . . . 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)
9	5, 7, 8	3bitri 300	. . . . 5 ⊢ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥[𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
10		albi 1826	. . . . 5 ⊢ (∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → (∀𝑥[𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ∀𝑥𝜑))
11	9, 10	syl5bb 286	. . . 4 ⊢ (∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
12	11	2alimi 1820	. . 3 ⊢ (∀𝑎∀𝑏∀𝑥([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
13	1, 3, 12	3syl 18	. 2 ⊢ (∀𝑥∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) → ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
14		df-ich 44571	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
15	14	albii 1827	. 2 ⊢ (∀𝑥[𝑎⇄𝑏]𝜑 ↔ ∀𝑥∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
16		df-ich 44571	. 2 ⊢ ([𝑎⇄𝑏]∀𝑥𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]∀𝑥𝜑 ↔ ∀𝑥𝜑))
17	13, 15, 16	3imtr4i 295	1 ⊢ (∀𝑥[𝑎⇄𝑏]𝜑 → [𝑎⇄𝑏]∀𝑥𝜑)

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1541  [wsb 2070  [wich 44570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-11 2158

This theorem depends on definitions:  df-bi 210  df-ex 1788  df-sb 2071  df-ich 44571

This theorem is referenced by: (None)