Theorem bj-substax12 37382

Description: Equivalent form of the axiom of substitution bj-ax12 37312. Although both sides need a DV condition on 𝑥, 𝑡 (or as in bj-ax12v3 37343 on 𝑡, 𝜑) to hold, their equivalence holds without DV conditions. The forward implication is proved in modal (K4) while the reverse implication is proved in modal (T5). The LHS has the advantage of not involving nested quantifiers on the same variable. Its metaweakening is proved from the core axiom schemes in bj-substw 37383. Note that in the LHS, the reverse implication holds by equs4 2450 (or equs4v 2033 if a DV condition is added on 𝑥, 𝑡 as in bj-ax12 37312), and the forward implication is sbalex 2281.

The LHS can be read as saying that if there exists a variable equal to a given term witnessing a given formula, then all variables equal to that term also witness that formula. The equivalent form of the LHS using only primitive symbols is (∀𝑥(𝑥 = 𝑡 → 𝜑) ∨ ∀𝑥(𝑥 = 𝑡 → ¬ 𝜑)), which expresses that a given formula is true at all variables equal to a given term, or false at all these variables. An equivalent form of the LHS using only the existential quantifier is ¬ (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) ∧ ∃𝑥(𝑥 = 𝑡 ∧ ¬ 𝜑)), which expresses that there can be no two variables both equal to a given term, one witnessing a formula and the other witnessing its negation. These equivalences do not hold in intuitionistic logic. The LHS should be the preferred form, and has the advantage of having no negation nor nested quantifiers. (Contributed by BJ, 21-May-2024.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-substax12	⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))

Proof of Theorem bj-substax12

Step	Hyp	Ref	Expression
1		bj-modal4 37374	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))
2	1	imim2i 17	. . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)))
3		19.38 1872	. . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
4	2, 3	syl 18	. . 3 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
5		hbe1a 2182	. . . . . 6 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
6	5, 1	syl 18	. . . . 5 ⊢ (∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))
7		bj-exlimg 37261	. . . . 5 ⊢ ((∃𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))))
8	6, 7	ax-mp 5	. . . 4 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)))
9		sp 2222	. . . . 5 ⊢ (∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑))
10	9	imim2i 17	. . . 4 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
11	8, 10	syl 18	. . 3 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) → (∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
12	4, 11	impbii 212	. 2 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)))
13		impexp 456	. . 3 ⊢ (((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ (𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
14	13	albii 1852	. 2 ⊢ (∀𝑥((𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))
15	12, 14	bitri 278	1 ⊢ ((∃𝑥(𝑥 = 𝑡 ∧ 𝜑) → ∀𝑥(𝑥 = 𝑡 → 𝜑)) ↔ ∀𝑥(𝑥 = 𝑡 → (𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568  ∃wex 1812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-10 2179  ax-12 2216

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813

This theorem is used by: (None)