Theorem ichan 47815

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 47806 instead of dfich2 47818 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 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		pm4.38 638	. . . . 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 1818	. . 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 1818	. 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 47806	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 47806	. . 3 ⊢ ([𝑎⇄𝑏]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓))
14	12, 13	anbi12i 629	. 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 47806	. 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 1540  [wsb 2068  [wich 47805

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 1912  ax-6 1969  ax-7 2010

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782  df-sb 2069  df-ich 47806

This theorem is referenced by:  ichim  47817