Theorem ichn 45997

Description: Negation does not affect interchangeability. (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 319	. . . 4 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
2		sbn 2277	. . . . . . . . 9 ⊢ ([𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑢 / 𝑏]𝜑)
3	2	sbbii 2080	. . . . . . . 8 ⊢ ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑)
4		sbn 2277	. . . . . . . 8 ⊢ ([𝑏 / 𝑎] ¬ [𝑢 / 𝑏]𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
5	3, 4	bitri 275	. . . . . . 7 ⊢ ([𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
6	5	sbbii 2080	. . . . . 6 ⊢ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ [𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
7		sbn 2277	. . . . . 6 ⊢ ([𝑎 / 𝑢] ¬ [𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
8	6, 7	bitri 275	. . . . 5 ⊢ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑)
9	8	bibi1i 339	. . . 4 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑) ↔ (¬ [𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ ¬ 𝜑))
10	1, 9	bitr4i 278	. . 3 ⊢ (([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ ([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
11	10	2albii 1823	. 2 ⊢ (∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑) ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
12		df-ich 45987	. 2 ⊢ ([𝑎⇄𝑏]𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏]𝜑 ↔ 𝜑))
13		df-ich 45987	. 2 ⊢ ([𝑎⇄𝑏] ¬ 𝜑 ↔ ∀𝑎∀𝑏([𝑎 / 𝑢][𝑏 / 𝑎][𝑢 / 𝑏] ¬ 𝜑 ↔ ¬ 𝜑))
14	11, 12, 13	3bitr4i 303	1 ⊢ ([𝑎⇄𝑏]𝜑 ↔ [𝑎⇄𝑏] ¬ 𝜑)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1540  [wsb 2068  [wich 45986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-10 2138  ax-12 2172

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-ex 1783  df-nf 1787  df-sb 2069  df-ich 45987

This theorem is referenced by:  ichim  45998