Theorem bj-alextruim 37207

Description: An equivalent expression for universal quantification over a non-occurring variable proved over ax-1 6-- ax-5 1938. The forward implication can be strengthened when ax-6 1995 is posited (which implies that models are non-empty), see spvw 2009. The reverse implication can be seen as a strengthening of ax-5 1938 (since the antecedent of the implication is weakened). See bj-exextruan 37208 for a dual statement.

An approximate meaning is: the universal quantification of a proposition over a non-occurring variable holds if and only if the proposition holds in nonempty universes. (Contributed by BJ, 14-Mar-2026.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-alextruim	⊢ (∀𝑥𝜑 ↔ (∃𝑥⊤ → 𝜑))

Distinct variable group:   𝜑,𝑥

Proof of Theorem bj-alextruim

Step	Hyp	Ref	Expression
1		bj-spvw 37205	. . 3 ⊢ (∃𝑥⊤ → (𝜑 ↔ ∀𝑥𝜑))
2	1	biimprcd 253	. 2 ⊢ (∀𝑥𝜑 → (∃𝑥⊤ → 𝜑))
3		ax-5 1938	. . . 4 ⊢ (𝜑 → ∀𝑥𝜑)
4	3	imim2i 17	. . 3 ⊢ ((∃𝑥⊤ → 𝜑) → (∃𝑥⊤ → ∀𝑥𝜑))
5		19.38 1867	. . . 4 ⊢ ((∃𝑥⊤ → ∀𝑥𝜑) → ∀𝑥(⊤ → 𝜑))
6		pm2.27 43	. . . . 5 ⊢ (⊤ → ((⊤ → 𝜑) → 𝜑))
7	6	mptru 1575	. . . 4 ⊢ ((⊤ → 𝜑) → 𝜑)
8	5, 7	sylg 1851	. . 3 ⊢ ((∃𝑥⊤ → ∀𝑥𝜑) → ∀𝑥𝜑)
9	4, 8	syl 18	. 2 ⊢ ((∃𝑥⊤ → 𝜑) → ∀𝑥𝜑)
10	2, 9	impbii 212	1 ⊢ (∀𝑥𝜑 ↔ (∃𝑥⊤ → 𝜑))

Syntax hints:   → wi 4   ↔ wb 209  ∀wal 1566  ⊤wtru 1569  ∃wex 1807

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938

This theorem depends on definitions:  df-bi 210  df-tru 1571  df-ex 1808

This theorem is referenced by:  bj-axseprep  37659