Theorem cbvals 17054

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 234	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 → 𝜓) ↔ (𝜒 → 𝜃)))
4	3	cbvalvw 1975	. . 3 ⊢ (∀𝑥(𝜑 → 𝜓) ↔ ∀𝑦(𝜒 → 𝜃))
5	1	cbvexvw 1976	. . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜒)
6	4, 5	anbi12i 464	. 2 ⊢ ((∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑) ↔ (∀𝑦(𝜒 → 𝜃) ∧ ∃𝑦𝜒))
7		df-als 17036	. 2 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ (∀𝑥(𝜑 → 𝜓) ∧ ∃𝑥𝜑))
8		df-als 17036	. 2 ⊢ (∀∃𝑦(𝜒 → 𝜃) ↔ (∀𝑦(𝜒 → 𝜃) ∧ ∃𝑦𝜒))
9	6, 7, 8	3bitr4i 212	1 ⊢ (∀∃𝑥(𝜑 → 𝜓) ↔ ∀∃𝑦(𝜒 → 𝜃))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1400  ∃wex 1545  ∀∃wals 17034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-als 17036

This theorem is referenced by: (None)