Theorem cbvals 50463

Description: Rule used to change bound variables, using implicit substitution. (Contributed by David A. Wheeler, 12-Jul-2026.)

Hypotheses

Ref	Expression
cbvals.1	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
cbvals.2	⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))

Assertion

Ref	Expression
cbvals	⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))

Distinct variable groups:   𝑥,𝑦   𝜒,𝑥   𝜃,𝑥   𝜑,𝑦   𝜓,𝑦

Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝜒(𝑦)   𝜃(𝑦)

Proof of Theorem cbvals

Step	Hyp	Ref	Expression
1		cbvals.1	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜒))
2		cbvals.2	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ 𝜃))
3	1, 2	imbi12d 347	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃)))
4	3	cbvalvw 2063	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜒 → 𝜃))
5	1	cbvexvw 2064	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜒)
6	4, 5	anbi12i 639	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ↔ (∀𝑦(𝜒 → 𝜃) ∧ ∃𝑦𝜒))
7		df-als 50446	. 2 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
8		df-als 50446	. 2 ⊢ (∀∃𝑦(𝜒 → 𝜃) ↔ (∀𝑦(𝜒 → 𝜃) ∧ ∃𝑦𝜒))
9	6, 7, 8	3bitr4i 306	1 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400  ∀wal 1565  ∃wex 1806  ∀∃wals 50444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035

This theorem depends on definitions:  df-bi 210  df-an 401  df-ex 1807  df-als 50446

This theorem is referenced by: (None)