Theorem rext0 45647

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 4458	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝐴 = ∅)
4	3	notbii 323	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ 𝐴 = ∅)
5		dfrex2 3092	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
6		df-ne 2959	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	4, 5, 6	3bitr4i 306	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 209   = wceq 1570   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  ∅c0 4286

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-ne 2959  df-ral 3080  df-rex 3090  df-dif 3908  df-nul 4287

This theorem is referenced by:  n0abso  45685