Theorem nfdOLD 2193

Description: Obsolete proof of nf5d 2118 as of 6-Oct-2021. (Contributed by Mario Carneiro, 24-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfdOLD.1	⊢ Ⅎ𝑥𝜑
nfdOLD.2	⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))

Assertion

Ref	Expression
nfdOLD	⊢ (𝜑 → Ⅎ𝑥𝜓)

Proof of Theorem nfdOLD

Step	Hyp	Ref	Expression
1		nfdOLD.1	. . 3 ⊢ Ⅎ𝑥𝜑
2		nfdOLD.2	. . 3 ⊢ (𝜑 → (𝜓 → ∀𝑥𝜓))
3	1, 2	alrimiOLD 2192	. 2 ⊢ (𝜑 → ∀𝑥(𝜓 → ∀𝑥𝜓))
4		df-nfOLD 1721	. 2 ⊢ (Ⅎ𝑥𝜓 ↔ ∀𝑥(𝜓 → ∀𝑥𝜓))
5	3, 4	sylibr 224	1 ⊢ (𝜑 → Ⅎ𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1481  ℲwnfOLD 1709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-12 2047

This theorem depends on definitions:  df-bi 197  df-ex 1705  df-nfOLD 1721

This theorem is referenced by:  nfdhOLD  2194  nfntOLD  2209