Theorem nfan1OLD 2224

Description: Obsolete proof of nfan1 2056 as of 6-Oct-2021. (Contributed by Mario Carneiro, 3-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfan1OLD.1	⊢ Ⅎ𝑥𝜑
nfan1OLD.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfan1OLD	⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Proof of Theorem nfan1OLD

Step	Hyp	Ref	Expression
1		nfan1OLD.2	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
2	1	nfrdOLD 2178	. . . 4 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	2	imdistani 722	. . 3 ⊢ ((𝜑 ∧ 𝜓) → (𝜑 ∧ ∀𝑥𝜓))
4		nfan1OLD.1	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	19.28OLD 2223	. . 3 ⊢ (∀𝑥(𝜑 ∧ 𝜓) ↔ (𝜑 ∧ ∀𝑥𝜓))
6	3, 5	sylibr 223	. 2 ⊢ ((𝜑 ∧ 𝜓) → ∀𝑥(𝜑 ∧ 𝜓))
7	6	nfiOLD 1725	1 ⊢ Ⅎ𝑥(𝜑 ∧ 𝜓)

Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ℲwnfOLD 1700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034

This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nfOLD 1712

This theorem is referenced by:  nfanOLDOLD  2225