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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 32608 | . . 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 4804 | . . . 4 ⊢ (𝜑 → 〈𝑀, 𝐴〉 = 〈𝑁, 𝐵〉) |
6 | 5 | opeq2d 4803 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = 〈2o, 〈𝑁, 𝐵〉〉) |
7 | df-goal 32608 | . . . . 5 ⊢ ∀𝑔𝑁𝐵 = 〈2o, 〈𝑁, 𝐵〉〉 | |
8 | 7 | eqcomi 2829 | . . . 4 ⊢ 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵 |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵) |
10 | 6, 9 | eqtrd 2855 | . 2 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = ∀𝑔𝑁𝐵) |
11 | 2, 10 | eqtrd 2855 | 1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 〈cop 4566 2oc2o 8089 ∀𝑔cgol 32601 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-goal 32608 |
This theorem is referenced by: satfv1 32629 satfdmlem 32634 fmlasuc 32652 fmla1 32653 satffunlem1lem1 32668 satffunlem1lem2 32669 satffunlem2lem1 32670 satffunlem2lem2 32672 satfv1fvfmla1 32689 2goelgoanfmla1 32690 |
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