Theorem rext0 45365

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 4437	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ 𝐴 = ∅)
4	3	notbii 320	. 2 ⊢ (¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑 ↔ ¬ 𝐴 = ∅)
5		dfrex2 3064	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ ¬ ∀𝑥 ∈ 𝐴 ¬ 𝜑)
6		df-ne 2933	. 2 ⊢ (𝐴 ≠ ∅ ↔ ¬ 𝐴 = ∅)
7	4, 5, 6	3bitr4i 303	1 ⊢ (∃𝑥 ∈ 𝐴 𝜑 ↔ 𝐴 ≠ ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1542   ≠ wne 2932  ∀wral 3051  ∃wrex 3061  ∅c0 4273

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-ne 2933  df-ral 3052  df-rex 3062  df-dif 3892  df-nul 4274

This theorem is referenced by:  n0abso  45403