Theorem bj-cbval 33977

Description: Changing a bound variable (universal quantification case) in a weak axiomatization, assuming that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbval.denote	⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
bj-cbval.denote2	⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
bj-cbval.maj	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
bj-cbval.equcomiv	⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)

Assertion

Ref	Expression
bj-cbval	⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

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

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

Proof of Theorem bj-cbval

Step	Hyp	Ref	Expression
1		bj-cbval.maj	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
2	1	biimpd 231	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3		bj-cbval.denote	. . 3 ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
4	2, 3	bj-cbvalimi 33975	. 2 ⊢ (∀𝑥𝜑 → ∀𝑦𝜓)
5		bj-cbval.equcomiv	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
6	1	biimprd 250	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
7	5, 6	syl 17	. . 3 ⊢ (𝑦 = 𝑥 → (𝜓 → 𝜑))
8		bj-cbval.denote2	. . 3 ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
9	7, 8	bj-cbvalimi 33975	. 2 ⊢ (∀𝑦𝜓 → ∀𝑥𝜑)
10	4, 9	impbii 211	1 ⊢ (∀𝑥𝜑 ↔ ∀𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 208  ∀wal 1531  ∃wex 1776

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907

This theorem depends on definitions:  df-bi 209  df-ex 1777

This theorem is referenced by: (None)