| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | supeq123d.b | . . . 4
⊢ (𝜑 → 𝐵 = 𝐸) | 
| 2 |  | supeq123d.a | . . . . . 6
⊢ (𝜑 → 𝐴 = 𝐷) | 
| 3 |  | supeq123d.c | . . . . . . . 8
⊢ (𝜑 → 𝐶 = 𝐹) | 
| 4 | 3 | breqd 5153 | . . . . . . 7
⊢ (𝜑 → (𝑥𝐶𝑦 ↔ 𝑥𝐹𝑦)) | 
| 5 | 4 | notbid 318 | . . . . . 6
⊢ (𝜑 → (¬ 𝑥𝐶𝑦 ↔ ¬ 𝑥𝐹𝑦)) | 
| 6 | 2, 5 | raleqbidv 3345 | . . . . 5
⊢ (𝜑 → (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ↔ ∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦)) | 
| 7 | 3 | breqd 5153 | . . . . . . 7
⊢ (𝜑 → (𝑦𝐶𝑥 ↔ 𝑦𝐹𝑥)) | 
| 8 | 3 | breqd 5153 | . . . . . . . 8
⊢ (𝜑 → (𝑦𝐶𝑧 ↔ 𝑦𝐹𝑧)) | 
| 9 | 2, 8 | rexeqbidv 3346 | . . . . . . 7
⊢ (𝜑 → (∃𝑧 ∈ 𝐴 𝑦𝐶𝑧 ↔ ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)) | 
| 10 | 7, 9 | imbi12d 344 | . . . . . 6
⊢ (𝜑 → ((𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))) | 
| 11 | 1, 10 | raleqbidv 3345 | . . . . 5
⊢ (𝜑 → (∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧) ↔ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))) | 
| 12 | 6, 11 | anbi12d 632 | . . . 4
⊢ (𝜑 → ((∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧)) ↔ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧)))) | 
| 13 | 1, 12 | rabeqbidv 3454 | . . 3
⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))} = {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}) | 
| 14 | 13 | unieqd 4919 | . 2
⊢ (𝜑 → ∪ {𝑥
∈ 𝐵 ∣
(∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))} = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))}) | 
| 15 |  | df-sup 9483 | . 2
⊢ sup(𝐴, 𝐵, 𝐶) = ∪ {𝑥 ∈ 𝐵 ∣ (∀𝑦 ∈ 𝐴 ¬ 𝑥𝐶𝑦 ∧ ∀𝑦 ∈ 𝐵 (𝑦𝐶𝑥 → ∃𝑧 ∈ 𝐴 𝑦𝐶𝑧))} | 
| 16 |  | df-sup 9483 | . 2
⊢ sup(𝐷, 𝐸, 𝐹) = ∪ {𝑥 ∈ 𝐸 ∣ (∀𝑦 ∈ 𝐷 ¬ 𝑥𝐹𝑦 ∧ ∀𝑦 ∈ 𝐸 (𝑦𝐹𝑥 → ∃𝑧 ∈ 𝐷 𝑦𝐹𝑧))} | 
| 17 | 14, 15, 16 | 3eqtr4g 2801 | 1
⊢ (𝜑 → sup(𝐴, 𝐵, 𝐶) = sup(𝐷, 𝐸, 𝐹)) |