Metamath Proof Explorer


Theorem 3rspcedvdw

Description: Triple application of rspcedvdw . (Contributed by SN, 20-Aug-2024)

Ref Expression
Hypotheses 3rspcedvdw.1 ( 𝑥 = 𝐴 → ( 𝜓𝜒 ) )
3rspcedvdw.2 ( 𝑦 = 𝐵 → ( 𝜒𝜃 ) )
3rspcedvdw.3 ( 𝑧 = 𝐶 → ( 𝜃𝜏 ) )
3rspcedvdw.a ( 𝜑𝐴𝑋 )
3rspcedvdw.b ( 𝜑𝐵𝑌 )
3rspcedvdw.c ( 𝜑𝐶𝑍 )
3rspcedvdw.4 ( 𝜑𝜏 )
Assertion 3rspcedvdw ( 𝜑 → ∃ 𝑥𝑋𝑦𝑌𝑧𝑍 𝜓 )

Proof

Step Hyp Ref Expression
1 3rspcedvdw.1 ( 𝑥 = 𝐴 → ( 𝜓𝜒 ) )
2 3rspcedvdw.2 ( 𝑦 = 𝐵 → ( 𝜒𝜃 ) )
3 3rspcedvdw.3 ( 𝑧 = 𝐶 → ( 𝜃𝜏 ) )
4 3rspcedvdw.a ( 𝜑𝐴𝑋 )
5 3rspcedvdw.b ( 𝜑𝐵𝑌 )
6 3rspcedvdw.c ( 𝜑𝐶𝑍 )
7 3rspcedvdw.4 ( 𝜑𝜏 )
8 1 2 3 rspc3ev ( ( ( 𝐴𝑋𝐵𝑌𝐶𝑍 ) ∧ 𝜏 ) → ∃ 𝑥𝑋𝑦𝑌𝑧𝑍 𝜓 )
9 4 5 6 7 8 syl31anc ( 𝜑 → ∃ 𝑥𝑋𝑦𝑌𝑧𝑍 𝜓 )