Theorem ichan 48093

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 48084 instead of dfich2 48096 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 2120	. . . . . . . 8 ⊢ ([𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
2	1	sbbii 2116	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
3	2	sbbii 2116	. . . . . 6 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
4		sban 2120	. . . . . . 7 ⊢ ([𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ ([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
5	4	sbbii 2116	. . . . . 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 2120	. . . . . 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 300	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
8		pm4.38 648	. . . . 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 286	. . . 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 1843	. . 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 1843	. 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 48084	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 48084	. . 3 ⊢ ([𝑎⇄𝑏]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓))
14	12, 13	anbi12i 639	. 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 48084	. 2 ⊢ ([𝑎⇄𝑏](𝜑 ∧ 𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
16	11, 14, 15	3imtr4i 295	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  [wsb 2097  [wich 48083

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836

This theorem depends on definitions:  df-bi 210  df-an 401  df-sb 2098  df-ich 48084

This theorem is referenced by:  ichim  48095