Theorem bj-cbvex 36960

Description: Changing a bound variable (existential 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 1912. (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-cbvex	⊢ (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-cbvex

Step	Hyp	Ref	Expression
1		bj-cbval.nf0	. . 3 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-cbval.nf1	. . 3 ⊢ (𝜑 → ∀𝑦𝜑)
3		ax5d 1913	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑦𝜓))
4		bj-cbval.denote2	. . . 4 ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
5	4	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑥∃𝑦 𝑦 = 𝑥)
6		bj-cbval.equcomiv	. . . . 5 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
7		bj-cbval.is	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜓 ↔ 𝜒))
8	6, 7	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑦 = 𝑥) → (𝜓 ↔ 𝜒))
9	8	biimpd 229	. . 3 ⊢ ((𝜑 ∧ 𝑦 = 𝑥) → (𝜓 → 𝜒))
10	1, 2, 3, 5, 9	bj-cbveximdv 36947	. 2 ⊢ (𝜑 → (∃𝑥𝜓 → ∃𝑦𝜒))
11		ax5d 1913	. . 3 ⊢ (𝜑 → (𝜒 → ∀𝑥𝜒))
12		bj-cbval.denote	. . . 4 ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
13	12	a1i 11	. . 3 ⊢ (𝜑 → ∀𝑦∃𝑥 𝑥 = 𝑦)
14	7	biimprd 248	. . 3 ⊢ ((𝜑 ∧ 𝑥 = 𝑦) → (𝜒 → 𝜓))
15	2, 1, 11, 13, 14	bj-cbveximdv 36947	. 2 ⊢ (𝜑 → (∃𝑦𝜒 → ∃𝑥𝜓))
16	10, 15	impbid 212	1 ⊢ (𝜑 → (∃𝑥𝜓 ↔ ∃𝑦𝜒))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1540  ∃wex 1781

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 1912

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1782

This theorem is referenced by: (None)