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Metamath Proof Explorer

Theorem 3rspcedvd

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

Ref Expression
Hypotheses 3rspcedvd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
3rspcedvd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
3rspcedvd.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )
3rspcedvd.1 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
3rspcedvd.2 ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝜒 ↔ 𝜃 ) )
3rspcedvd.3 ⊢ ( ( 𝜑 ∧ 𝑧 = 𝐶 ) → ( 𝜃 ↔ 𝜏 ) )
3rspcedvd.4 ⊢ ( 𝜑 → 𝜏 )
Assertion 3rspcedvd ( 𝜑 → ∃ 𝑥 ∈ 𝐷 ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜓 )

Proof

Step Hyp Ref Expression
1 3rspcedvd.a ⊢ ( 𝜑 → 𝐴 ∈ 𝐷 )
2 3rspcedvd.b ⊢ ( 𝜑 → 𝐵 ∈ 𝐷 )
3 3rspcedvd.c ⊢ ( 𝜑 → 𝐶 ∈ 𝐷 )
4 3rspcedvd.1 ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( 𝜓 ↔ 𝜒 ) )
5 3rspcedvd.2 ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝜒 ↔ 𝜃 ) )
6 3rspcedvd.3 ⊢ ( ( 𝜑 ∧ 𝑧 = 𝐶 ) → ( 𝜃 ↔ 𝜏 ) )
7 3rspcedvd.4 ⊢ ( 𝜑 → 𝜏 )
8 4 2rexbidv ⊢ ( ( 𝜑 ∧ 𝑥 = 𝐴 ) → ( ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜓 ↔ ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜒 ) )
9 5 rexbidv ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( ∃ 𝑧 ∈ 𝐷 𝜒 ↔ ∃ 𝑧 ∈ 𝐷 𝜃 ) )
10 3 6 7 rspcedvd ⊢ ( 𝜑 → ∃ 𝑧 ∈ 𝐷 𝜃 )
11 2 9 10 rspcedvd ⊢ ( 𝜑 → ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜒 )
12 1 8 11 rspcedvd ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐷 ∃ 𝑦 ∈ 𝐷 ∃ 𝑧 ∈ 𝐷 𝜓 )