Theorem bj-exextruan 36979

Description: An equivalent expression for existential quantification over a non-occurring variable proved over ax-1 6-- ax-5 1917. The forward implication can be seen as a strengthening of ax-5 1917 (a conjunct is added to the consequent of the implication). The reverse implication can be strengthened when ax-6 1974 is posited (which implies that models are non-empty), see 19.8v 1990. See bj-alextruim 36978 for a dual statement.

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

Assertion

Ref	Expression
bj-exextruan	⊢ (∃𝑥𝜑 ↔ (∃𝑥⊤ ∧ 𝜑))

Distinct variable group:   𝜑,𝑥

Proof of Theorem bj-exextruan

Step	Hyp	Ref	Expression
1		trud 1557	. . . 4 ⊢ (𝜑 → ⊤)
2	1	eximi 1842	. . 3 ⊢ (∃𝑥𝜑 → ∃𝑥⊤)
3		ax5e 1919	. . 3 ⊢ (∃𝑥𝜑 → 𝜑)
4	2, 3	jca 516	. 2 ⊢ (∃𝑥𝜑 → (∃𝑥⊤ ∧ 𝜑))
5		bj-spvew 36977	. . 3 ⊢ (∃𝑥⊤ → (𝜑 ↔ ∃𝑥𝜑))
6	5	biimpa 477	. 2 ⊢ ((∃𝑥⊤ ∧ 𝜑) → ∃𝑥𝜑)
7	4, 6	impbii 210	1 ⊢ (∃𝑥𝜑 ↔ (∃𝑥⊤ ∧ 𝜑))

Syntax hints:   ↔ wb 207   ∧ wa 396  ⊤wtru 1548  ∃wex 1786

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917

This theorem depends on definitions:  df-bi 208  df-an 397  df-tru 1550  df-ex 1787

This theorem is referenced by: (None)