Theorem ichan 44859

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 44850 instead of dfich2 44862 to reduce axioms. (Revised by SN, 4-May-2024.)

Assertion

Ref	Expression
ichan	⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))

Proof of Theorem ichan

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sban 2086	. . . . . . . 8 ⊢ ([𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
2	1	sbbii 2082	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
3	2	sbbii 2082	. . . . . 6 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
4		sban 2086	. . . . . . 7 ⊢ ([𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ ([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
5	4	sbbii 2082	. . . . . 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 2086	. . . . . 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))
7	3, 5, 6	3bitri 296	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
8		pm4.38 634	. . . . 5 ⊢ ((([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑) ∧ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓)) → (([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓) ↔ (𝜑 ∧ 𝜓)))
9	7, 8	syl5bb 282	. . . 4 ⊢ ((([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑) ∧ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓)) → ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
10	9	alanimi 1822	. . 3 ⊢ ((∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑) ∧ ∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓)) → ∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
11	10	alanimi 1822	. 2 ⊢ ((∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑) ∧ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓)) → ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
12		df-ich 44850	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 44850	. . 3 ⊢ ([𝑎⇄𝑏]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓))
14	12, 13	anbi12i 626	. 2 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) ↔ (∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑) ∧ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓)))
15		df-ich 44850	. 2 ⊢ ([𝑎⇄𝑏](𝜑 ∧ 𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
16	11, 14, 15	3imtr4i 291	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1539  [wsb 2070  [wich 44849

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815

This theorem depends on definitions:  df-bi 206  df-an 396  df-sb 2071  df-ich 44850

This theorem is referenced by:  ichim  44861