Theorem nfralwOLD 3303

Description: Obsolete version of nfralw 3302 as of 13-Dec-2024. (Contributed by NM, 1-Sep-1999.) (Revised by Gino Giotto, 10-Jan-2024.) (Proof modification is discouraged.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nfralw.1	⊢ Ⅎ𝑥𝐴
nfralw.2	⊢ Ⅎ𝑥𝜑

Assertion

Ref	Expression
nfralwOLD	⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐴(𝑥,𝑦)

Proof of Theorem nfralwOLD

Step	Hyp	Ref	Expression
1		nftru 1798	. . 3 ⊢ Ⅎ𝑦⊤
2		nfralw.1	. . . 4 ⊢ Ⅎ𝑥𝐴
3	2	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝐴)
4		nfralw.2	. . . 4 ⊢ Ⅎ𝑥𝜑
5	4	a1i 11	. . 3 ⊢ (⊤ → Ⅎ𝑥𝜑)
6	1, 3, 5	nfraldw 3300	. 2 ⊢ (⊤ → Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑)
7	6	mptru 1540	1 ⊢ Ⅎ𝑥∀𝑦 ∈ 𝐴 𝜑

Syntax hints:  ⊤wtru 1534  Ⅎwnf 1777  Ⅎwnfc 2877  ∀wral 3055

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-10 2129  ax-11 2146  ax-12 2163

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-nf 1778  df-clel 2804  df-nfc 2879  df-ral 3056

This theorem is referenced by: (None)