Theorem ichan 47380

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 47371 instead of dfich2 47383 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 2078	. . . . . . . 8 ⊢ ([𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
2	1	sbbii 2074	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
3	2	sbbii 2074	. . . . . 6 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
4		sban 2078	. . . . . . 7 ⊢ ([𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ ([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
5	4	sbbii 2074	. . . . . 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 2078	. . . . . 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 297	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
8		pm4.38 637	. . . . 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	bitrid 283	. . . 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 1813	. . 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 1813	. 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 47371	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 47371	. . 3 ⊢ ([𝑎⇄𝑏]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓))
14	12, 13	anbi12i 628	. 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 47371	. 2 ⊢ ([𝑎⇄𝑏](𝜑 ∧ 𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
16	11, 14, 15	3imtr4i 292	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1535  [wsb 2062  [wich 47370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806

This theorem depends on definitions:  df-bi 207  df-an 396  df-sb 2063  df-ich 47371

This theorem is referenced by:  ichim  47382