Theorem bj-cbval 37118

Description: Changing a bound variable (universal quantification case) in a weak axiomatization that assumes that all variables denote (which is valid in inclusive free logic) and that equality is symmetric. (Contributed by BJ, 12-Mar-2023.) Proved from ax-1 6-- ax-5 1930. (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-cbval.denote	⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
bj-cbval.denote2	⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
bj-cbval.equcomiv	⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
bj-cbval.nf0	⊢ (𝜑 → ∀𝑥𝜑)
bj-cbval.nf1	⊢ (𝜑 → ∀𝑦𝜑)
bj-cbval.is	⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
bj-cbval	⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

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

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

Proof of Theorem bj-cbval

Step	Hyp	Ref	Expression
1		bj-cbval.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbval.nf1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		ax5e 1932	. . . 4 ⊢ (∃𝑥𝜒 → 𝜒)
4	3	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑥𝜒 → 𝜒))
5		bj-cbval.denote	. . . 4 ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
6	5	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑥 𝑥 = 𝑦)
7		bj-cbval.is	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
8	7	biimpd 231	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 → 𝜒))
9	1, 2, 4, 6, 8	bj-cbvalimdv 37105	. 2 ⊢ (𝜑 → (∀𝑥𝜓 → ∀𝑦𝜒))
10		ax5e 1932	. . . 4 ⊢ (∃𝑦𝜓 → 𝜓)
11	10	a1i 11	. . 3 ⊢ (𝜑 → (∃𝑦𝜓 → 𝜓))
12		bj-cbval.denote2	. . . 4 ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
13	12	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥∃𝑦 𝑦 = 𝑥)
14		bj-cbval.equcomiv	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
15	7	biimprd 250	. . . 4 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜒 → 𝜓))
16	14, 15	sylan2 602	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑥) → (𝜒 → 𝜓))
17	2, 1, 11, 13, 16	bj-cbvalimdv 37105	. 2 ⊢ (𝜑 → (∀𝑦𝜒 → ∀𝑥𝜓))
18	9, 17	impbid 214	1 ⊢ (𝜑 → (∀𝑥𝜓 ↔ ∀𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558  ∃wex 1799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930

This theorem depends on definitions:  df-bi 209  df-an 400  df-ex 1800

This theorem is referenced by: (None)