This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem vtocl3gaOLD

Description: Obsolete version of vtocl3ga as of 31-May-2025. (Contributed by NM, 20-Aug-1995) Reduce axiom usage. (Revised by GG, 3-Oct-2024) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Hypotheses vtocl3ga.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
vtocl3ga.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
vtocl3ga.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
vtocl3ga.4 ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 )
Assertion vtocl3gaOLD ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )

Proof

Step Hyp Ref Expression
1 vtocl3ga.1 ⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2 vtocl3ga.2 ⊢ ( 𝑦 = 𝐵 → ( 𝜓 ↔ 𝜒 ) )
3 vtocl3ga.3 ⊢ ( 𝑧 = 𝐶 → ( 𝜒 ↔ 𝜃 ) )
4 vtocl3ga.4 ⊢ ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 )
5 eleq1 ⊢ ( 𝑥 = 𝐴 → ( 𝑥 ∈ 𝐷 ↔ 𝐴 ∈ 𝐷 ) )
6 5 3anbi1d ⊢ ( 𝑥 = 𝐴 → ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ) )
7 6 1 imbi12d ⊢ ( 𝑥 = 𝐴 → ( ( ( 𝑥 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜑 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜓 ) ) )
8 eleq1 ⊢ ( 𝑦 = 𝐵 → ( 𝑦 ∈ 𝑅 ↔ 𝐵 ∈ 𝑅 ) )
9 8 3anbi2d ⊢ ( 𝑦 = 𝐵 → ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ) )
10 9 2 imbi12d ⊢ ( 𝑦 = 𝐵 → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝑦 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜓 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜒 ) ) )
11 eleq1 ⊢ ( 𝑧 = 𝐶 → ( 𝑧 ∈ 𝑆 ↔ 𝐶 ∈ 𝑆 ) )
12 11 3anbi3d ⊢ ( 𝑧 = 𝐶 → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) ↔ ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) ) )
13 12 3 imbi12d ⊢ ( 𝑧 = 𝐶 → ( ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝑧 ∈ 𝑆 ) → 𝜒 ) ↔ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 ) ) )
14 7 10 13 4 vtocl3g ⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 ) )
15 14 pm2.43i ⊢ ( ( 𝐴 ∈ 𝐷 ∧ 𝐵 ∈ 𝑅 ∧ 𝐶 ∈ 𝑆 ) → 𝜃 )