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| Mirrors > Home > MPE Home > Th. List > Mathboxes > goaleq12d | Structured version Visualization version GIF version | ||
| Description: Equality of the "Godel-set of universal quantification". (Contributed by AV, 18-Sep-2023.) |
| Ref | Expression |
|---|---|
| goaleq12d.1 | ⊢ (𝜑 → 𝑀 = 𝑁) |
| goaleq12d.2 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| goaleq12d | ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-goal 35347 | . . 3 ⊢ ∀𝑔𝑀𝐴 = 〈2o, 〈𝑀, 𝐴〉〉 | |
| 2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = 〈2o, 〈𝑀, 𝐴〉〉) |
| 3 | goaleq12d.1 | . . . . 5 ⊢ (𝜑 → 𝑀 = 𝑁) | |
| 4 | goaleq12d.2 | . . . . 5 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 5 | 3, 4 | opeq12d 4881 | . . . 4 ⊢ (𝜑 → 〈𝑀, 𝐴〉 = 〈𝑁, 𝐵〉) |
| 6 | 5 | opeq2d 4880 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = 〈2o, 〈𝑁, 𝐵〉〉) |
| 7 | df-goal 35347 | . . . . 5 ⊢ ∀𝑔𝑁𝐵 = 〈2o, 〈𝑁, 𝐵〉〉 | |
| 8 | 7 | eqcomi 2746 | . . . 4 ⊢ 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵 |
| 9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵) |
| 10 | 6, 9 | eqtrd 2777 | . 2 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = ∀𝑔𝑁𝐵) |
| 11 | 2, 10 | eqtrd 2777 | 1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 〈cop 4632 2oc2o 8500 ∀𝑔cgol 35340 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-goal 35347 |
| This theorem is referenced by: satfv1 35368 satfdmlem 35373 fmlasuc 35391 fmla1 35392 satffunlem1lem1 35407 satffunlem1lem2 35408 satffunlem2lem1 35409 satffunlem2lem2 35411 satfv1fvfmla1 35428 2goelgoanfmla1 35429 |
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