Theorem ichbidv

Description: Formula building rule for interchangeability (deduction). (Contributed by SN, 4-May-2024)

	Ref	Expression
Hypothesis	ichbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
Assertion	ichbidv	⊢ ( 𝜑 → ( [ 𝑥 ⇄ 𝑦 ] 𝜓 ↔ [ 𝑥 ⇄ 𝑦 ] 𝜒 ) )

Proof

Step	Hyp	Ref	Expression
1		ichbidv.1	⊢ ( 𝜑 → ( 𝜓 ↔ 𝜒 ) )
2	1	sbbidv	⊢ ( 𝜑 → ( [ 𝑎 / 𝑦 ] 𝜓 ↔ [ 𝑎 / 𝑦 ] 𝜒 ) )
3	2	sbbidv	⊢ ( 𝜑 → ( [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ) )
4	3	sbbidv	⊢ ( 𝜑 → ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ) )
5	4 1	bibi12d	⊢ ( 𝜑 → ( ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ 𝜓 ) ↔ ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ↔ 𝜒 ) ) )
6	5	albidv	⊢ ( 𝜑 → ( ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ 𝜓 ) ↔ ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ↔ 𝜒 ) ) )
7	6	albidv	⊢ ( 𝜑 → ( ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ 𝜓 ) ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ↔ 𝜒 ) ) )
8		df-ich	⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜓 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜓 ↔ 𝜓 ) )
9		df-ich	⊢ ( [ 𝑥 ⇄ 𝑦 ] 𝜒 ↔ ∀ 𝑥 ∀ 𝑦 ( [ 𝑥 / 𝑎 ] [ 𝑦 / 𝑥 ] [ 𝑎 / 𝑦 ] 𝜒 ↔ 𝜒 ) )
10	7 8 9	3bitr4g	⊢ ( 𝜑 → ( [ 𝑥 ⇄ 𝑦 ] 𝜓 ↔ [ 𝑥 ⇄ 𝑦 ] 𝜒 ) )