Theorem nfabdOLD 3004

Description: Obsolete version of nfabd 3001 as of 10-May-2023. (Contributed by Mario Carneiro, 8-Oct-2016.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfabdOLD.1	⊢ Ⅎ𝑦𝜑
nfabdOLD.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabdOLD	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Proof of Theorem nfabdOLD

Step	Hyp	Ref	Expression
1		nfabdOLD.1	. 2 ⊢ Ⅎ𝑦𝜑
2		nfabdOLD.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	2	adantr 483	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfabd2 3002	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1535  Ⅎwnf 1784  {cab 2799  Ⅎwnfc 2961

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-13 2390  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963

This theorem is referenced by: (None)