Theorem ichan 48061

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 48052 instead of dfich2 48064 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 2113	. . . . . . . 8 ⊢ ([𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
2	1	sbbii 2109	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
3	2	sbbii 2109	. . . . . 6 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ [𝑎 / 𝑥][𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓))
4		sban 2113	. . . . . . 7 ⊢ ([𝑏 / 𝑎]([𝑥 / 𝑏]𝜑 ∧ [𝑥 / 𝑏]𝜓) ↔ ([𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
5	4	sbbii 2109	. . . . . 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 2113	. . . . . 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 299	. . . . 5 ⊢ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ ([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ∧ [𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓))
8		pm4.38 646	. . . . 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 285	. . . 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 1836	. . 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 1836	. 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 48052	. . 3 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 48052	. . 3 ⊢ ([𝑎⇄𝑏]𝜓 ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏]𝜓 ↔ 𝜓))
14	12, 13	anbi12i 637	. 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 48052	. 2 ⊢ ([𝑎⇄𝑏](𝜑 ∧ 𝜓) ↔ ∀𝑎∀𝑏([𝑎 / 𝑥][𝑏 / 𝑎][𝑥 / 𝑏](𝜑 ∧ 𝜓) ↔ (𝜑 ∧ 𝜓)))
16	11, 14, 15	3imtr4i 294	1 ⊢ (([𝑎⇄𝑏]𝜑 ∧ [𝑎⇄𝑏]𝜓) → [𝑎⇄𝑏](𝜑 ∧ 𝜓))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  [wsb 2090  [wich 48051

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829

This theorem depends on definitions:  df-bi 209  df-an 400  df-sb 2091  df-ich 48052

This theorem is referenced by:  ichim  48063