Metamath Proof Explorer


Theorem ralxp3es

Description: Restricted for-all over a triple cross product with explicit substitution. (Contributed by Scott Fenton, 22-Aug-2024)

Ref Expression
Assertion ralxp3es ( ∀ 𝑥 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑 ↔ ∀ 𝑦𝐴𝑧𝐵𝑤𝐶 𝜑 )

Proof

Step Hyp Ref Expression
1 nfsbc1v 𝑦 [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
2 nfcv 𝑧 ( 1st ‘ ( 1st𝑥 ) )
3 nfsbc1v 𝑧 [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
4 2 3 nfsbcw 𝑧 [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
5 nfcv 𝑤 ( 1st ‘ ( 1st𝑥 ) )
6 nfcv 𝑤 ( 2nd ‘ ( 1st𝑥 ) )
7 nfsbc1v 𝑤 [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
8 6 7 nfsbcw 𝑤 [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
9 5 8 nfsbcw 𝑤 [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑
10 nfv 𝑥 𝜑
11 sbcoteq1a ( 𝑥 = ⟨ ⟨ 𝑦 , 𝑧 ⟩ , 𝑤 ⟩ → ( [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑𝜑 ) )
12 1 4 9 10 11 ralxp3f ( ∀ 𝑥 ∈ ( ( 𝐴 × 𝐵 ) × 𝐶 ) [ ( 1st ‘ ( 1st𝑥 ) ) / 𝑦 ] [ ( 2nd ‘ ( 1st𝑥 ) ) / 𝑧 ] [ ( 2nd𝑥 ) / 𝑤 ] 𝜑 ↔ ∀ 𝑦𝐴𝑧𝐵𝑤𝐶 𝜑 )