Metamath Proof Explorer


Theorem 3rspcedvd

Description: Triple application of rspcedvd . (Contributed by Steven Nguyen, 27-Feb-2023)

Ref Expression
Hypotheses 3rspcedvd.a ( 𝜑𝐴𝐷 )
3rspcedvd.b ( 𝜑𝐵𝐷 )
3rspcedvd.c ( 𝜑𝐶𝐷 )
3rspcedvd.1 ( ( 𝜑𝑥 = 𝐴 ) → ( 𝜓𝜒 ) )
3rspcedvd.2 ( ( 𝜑𝑦 = 𝐵 ) → ( 𝜒𝜃 ) )
3rspcedvd.3 ( ( 𝜑𝑧 = 𝐶 ) → ( 𝜃𝜏 ) )
3rspcedvd.4 ( 𝜑𝜏 )
Assertion 3rspcedvd ( 𝜑 → ∃ 𝑥𝐷𝑦𝐷𝑧𝐷 𝜓 )

Proof

Step Hyp Ref Expression
1 3rspcedvd.a ( 𝜑𝐴𝐷 )
2 3rspcedvd.b ( 𝜑𝐵𝐷 )
3 3rspcedvd.c ( 𝜑𝐶𝐷 )
4 3rspcedvd.1 ( ( 𝜑𝑥 = 𝐴 ) → ( 𝜓𝜒 ) )
5 3rspcedvd.2 ( ( 𝜑𝑦 = 𝐵 ) → ( 𝜒𝜃 ) )
6 3rspcedvd.3 ( ( 𝜑𝑧 = 𝐶 ) → ( 𝜃𝜏 ) )
7 3rspcedvd.4 ( 𝜑𝜏 )
8 4 2rexbidv ( ( 𝜑𝑥 = 𝐴 ) → ( ∃ 𝑦𝐷𝑧𝐷 𝜓 ↔ ∃ 𝑦𝐷𝑧𝐷 𝜒 ) )
9 5 rexbidv ( ( 𝜑𝑦 = 𝐵 ) → ( ∃ 𝑧𝐷 𝜒 ↔ ∃ 𝑧𝐷 𝜃 ) )
10 3 6 7 rspcedvd ( 𝜑 → ∃ 𝑧𝐷 𝜃 )
11 2 9 10 rspcedvd ( 𝜑 → ∃ 𝑦𝐷𝑧𝐷 𝜒 )
12 1 8 11 rspcedvd ( 𝜑 → ∃ 𝑥𝐷𝑦𝐷𝑧𝐷 𝜓 )