Theorem ichn 43675

Description: Negation does not affect interchangability. (Contributed by SN, 30-Aug-2023.)

Assertion

Ref	Expression
ichn	⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑)

Proof of Theorem ichn

Dummy variable 𝑢 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		notbi 321	. . . 4 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
2		sbn 2287	. . . . . . . . 9 ⊢ ([𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑢 / 𝑏]𝜑)
3	2	sbbii 2081	. . . . . . . 8 ⊢ ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑)
4		sbn 2287	. . . . . . . 8 ⊢ ([𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
5	3, 4	bitri 277	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
6	5	sbbii 2081	. . . . . 6 ⊢ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
7		sbn 2287	. . . . . 6 ⊢ ([𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
8	6, 7	bitri 277	. . . . 5 ⊢ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
9	8	bibi1i 341	. . . 4 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
10	1, 9	bitr4i 280	. . 3 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
11	10	2albii 1821	. 2 ⊢ (∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
12		df-ich 43655	. 2 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 43655	. 2 ⊢ ([𝑎⇄𝑏] ¬ 𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
14	11, 12, 13	3bitr4i 305	1 ⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 208  ∀wal 1535  [wsb 2069  [wich 43654

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-10 2145  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1781  df-nf 1785  df-sb 2070  df-ich 43655

This theorem is referenced by: (None)