Theorem bj-cbvex 34831

Description: Changing a bound variable (existential 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-cbvex	⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

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

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

Proof of Theorem bj-cbvex

Step	Hyp	Ref	Expression
1		bj-cbval.equcomiv	. . . 4 ⊢ (𝑦 = 𝑥 → 𝑥 = 𝑦)
2		bj-cbval.maj	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	2	biimpd 228	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	1, 3	syl 17	. . 3 ⊢ (𝑦 = 𝑥 → (𝜑 → 𝜓))
5		bj-cbval.denote2	. . 3 ⊢ ∀𝑥∃𝑦 𝑦 = 𝑥
6	4, 5	bj-cbveximi 34829	. 2 ⊢ (∃𝑥𝜑 → ∃𝑦𝜓)
7	2	biimprd 247	. . 3 ⊢ (𝑥 = 𝑦 → (𝜓 → 𝜑))
8		bj-cbval.denote	. . 3 ⊢ ∀𝑦∃𝑥 𝑥 = 𝑦
9	7, 8	bj-cbveximi 34829	. 2 ⊢ (∃𝑦𝜓 → ∃𝑥𝜑)
10	6, 9	impbii 208	1 ⊢ (∃𝑥𝜑 ↔ ∃𝑦𝜓)

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537  ∃wex 1782

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913

This theorem depends on definitions:  df-bi 206  df-ex 1783

This theorem is referenced by: (None)