Theorem ichbidv 43970

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

Hypothesis

Ref	Expression
ichbidv.1	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝜑,𝑥   𝜑,𝑦

Allowed substitution hints:   𝜓(𝑥,𝑦)   𝜒(𝑥,𝑦)

Proof of Theorem ichbidv

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ichbidv.1	. . . . . . . 8 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
2	1	sbbidv 2084	. . . . . . 7 ⊢ (𝜑 → ([𝑎 / 𝑦]𝜓 ↔ [𝑎 / 𝑦]𝜒))
3	2	sbbidv 2084	. . . . . 6 ⊢ (𝜑 → ([𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑦 / 𝑥][𝑎 / 𝑦]𝜒))
4	3	sbbidv 2084	. . . . 5 ⊢ (𝜑 → ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ [𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒))
5	4, 1	bibi12d 349	. . . 4 ⊢ (𝜑 → (([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒)))
6	5	albidv 1921	. . 3 ⊢ (𝜑 → (∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒)))
7	6	albidv 1921	. 2 ⊢ (𝜑 → (∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓) ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒)))
8		df-ich 43963	. 2 ⊢ ([𝑥⇄𝑦]𝜓 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜓 ↔ 𝜓))
9		df-ich 43963	. 2 ⊢ ([𝑥⇄𝑦]𝜒 ↔ ∀𝑥∀𝑦([𝑥 / 𝑎][𝑦 / 𝑥][𝑎 / 𝑦]𝜒 ↔ 𝜒))
10	7, 8, 9	3bitr4g 317	1 ⊢ (𝜑 → ([𝑥⇄𝑦]𝜓 ↔ [𝑥⇄𝑦]𝜒))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1536  [wsb 2069  [wich 43962

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 1911

This theorem depends on definitions:  df-bi 210  df-sb 2070  df-ich 43963

This theorem is referenced by:  ichim  43974