Theorem rext0 45397

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 143	. . . 4 ⊢ ¬ ¬ 𝜑
3	2	ralf0 4428	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝐴 = ∅)
4	3	notbii 322	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ 𝐴 = ∅)
5		dfrex2 3068	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
6		df-ne 2937	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	4, 5, 6	3bitr4i 305	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 208   = wceq 1548   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  ∅c0 4264

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-ne 2937  df-ral 3056  df-rex 3066  df-dif 3888  df-nul 4265

This theorem is referenced by:  n0abso  45435