Theorem r19.28zf 45685

Description: Restricted quantifier version of Theorem 19.28 of [Margaris] p. 90. It is valid only when the domain of quantification is not empty. (Contributed by Glauco Siliprandi, 24-Jan-2025.)

Hypotheses

Ref	Expression
r19.28zf.1	⊢ Ⅎ𝑥𝜑
r19.28zf.2	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
r19.28zf	⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Proof of Theorem r19.28zf

Step	Hyp	Ref	Expression
1		r19.26 3116	. 2 ⊢ (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓))
2		r19.28zf.1	. . . 4 ⊢ Ⅎ𝑥𝜑
3		r19.28zf.2	. . . 4 ⊢ Ⅎ𝑥𝐴
4	2, 3	r19.3rzf 45684	. . 3 ⊢ (𝐴 ≠ ∅ → (𝜑 ↔ ∀𝑥 ∈ 𝐴 𝜑))
5	4	anbi1d 639	. 2 ⊢ (𝐴 ≠ ∅ → ((𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓) ↔ (∀𝑥 ∈ 𝐴 𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))
6	1, 5	bitr4id 292	1 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 (𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥 ∈ 𝐴 𝜓)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398  Ⅎwnf 1797  Ⅎwnfc 2903   ≠ wne 2951  ∀wral 3070  ∅c0 4280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981  ax-7 2022  ax-8 2138  ax-9 2146  ax-11 2185  ax-12 2206  ax-ext 2728

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 857  df-tru 1557  df-fal 1567  df-ex 1794  df-nf 1798  df-sb 2085  df-clab 2735  df-cleq 2748  df-clel 2831  df-nfc 2905  df-ne 2952  df-ral 3071  df-dif 3902  df-nul 4281

This theorem is referenced by:  iindif2f  45686