Theorem rext0 45538

Description: Nonempty existential quantification of a theorem is true. (Contributed by Eric Schmidt, 19-Oct-2025.)

Hypothesis

Ref	Expression
rext0.1	⊢ 𝜑

Assertion

Ref	Expression
rext0	⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem rext0

Step	Hyp	Ref	Expression
1		rext0.1	. . . . 5 ⊢ 𝜑
2	1	notnoti 144	. . . 4 ⊢ ¬ ¬ 𝜑
3	2	ralf0 4463	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝐴 = ∅)
4	3	notbii 323	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ 𝐴 = ∅)
5		dfrex2 3098	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
6		df-ne 2965	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	4, 5, 6	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1567   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  ∅c0 4294

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-ne 2965  df-ral 3086  df-rex 3096  df-dif 3916  df-nul 4295

This theorem is referenced by:  n0abso  45576