Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
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 32589 | . . 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 4811 | . . . 4 ⊢ (𝜑 → 〈𝑀, 𝐴〉 = 〈𝑁, 𝐵〉) |
6 | 5 | opeq2d 4810 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = 〈2o, 〈𝑁, 𝐵〉〉) |
7 | df-goal 32589 | . . . . 5 ⊢ ∀𝑔𝑁𝐵 = 〈2o, 〈𝑁, 𝐵〉〉 | |
8 | 7 | eqcomi 2830 | . . . 4 ⊢ 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵 |
9 | 8 | a1i 11 | . . 3 ⊢ (𝜑 → 〈2o, 〈𝑁, 𝐵〉〉 = ∀𝑔𝑁𝐵) |
10 | 6, 9 | eqtrd 2856 | . 2 ⊢ (𝜑 → 〈2o, 〈𝑀, 𝐴〉〉 = ∀𝑔𝑁𝐵) |
11 | 2, 10 | eqtrd 2856 | 1 ⊢ (𝜑 → ∀𝑔𝑀𝐴 = ∀𝑔𝑁𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 〈cop 4573 2oc2o 8096 ∀𝑔cgol 32582 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-goal 32589 |
This theorem is referenced by: satfv1 32610 satfdmlem 32615 fmlasuc 32633 fmla1 32634 satffunlem1lem1 32649 satffunlem1lem2 32650 satffunlem2lem1 32651 satffunlem2lem2 32653 satfv1fvfmla1 32670 2goelgoanfmla1 32671 |
Copyright terms: Public domain | W3C validator |