Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > eqbrrdv | Structured version Visualization version GIF version |
Description: Deduction from extensionality principle for relations. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
eqbrrdv.1 | ⊢ (𝜑 → Rel 𝐴) |
eqbrrdv.2 | ⊢ (𝜑 → Rel 𝐵) |
eqbrrdv.3 | ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) |
Ref | Expression |
---|---|
eqbrrdv | ⊢ (𝜑 → 𝐴 = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqbrrdv.3 | . . . 4 ⊢ (𝜑 → (𝑥𝐴𝑦 ↔ 𝑥𝐵𝑦)) | |
2 | df-br 5069 | . . . 4 ⊢ (𝑥𝐴𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐴) | |
3 | df-br 5069 | . . . 4 ⊢ (𝑥𝐵𝑦 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵) | |
4 | 1, 2, 3 | 3bitr3g 315 | . . 3 ⊢ (𝜑 → (〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
5 | 4 | alrimivv 1929 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵)) |
6 | eqbrrdv.1 | . . 3 ⊢ (𝜑 → Rel 𝐴) | |
7 | eqbrrdv.2 | . . 3 ⊢ (𝜑 → Rel 𝐵) | |
8 | eqrel 5660 | . . 3 ⊢ ((Rel 𝐴 ∧ Rel 𝐵) → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) | |
9 | 6, 7, 8 | syl2anc 586 | . 2 ⊢ (𝜑 → (𝐴 = 𝐵 ↔ ∀𝑥∀𝑦(〈𝑥, 𝑦〉 ∈ 𝐴 ↔ 〈𝑥, 𝑦〉 ∈ 𝐵))) |
10 | 5, 9 | mpbird 259 | 1 ⊢ (𝜑 → 𝐴 = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∀wal 1535 = wceq 1537 ∈ wcel 2114 〈cop 4575 class class class wbr 5068 Rel wrel 5562 |
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 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-in 3945 df-ss 3954 df-br 5069 df-opab 5131 df-xp 5563 df-rel 5564 |
This theorem is referenced by: eqbrrdva 5742 oppcsect2 17051 qusxpid 30930 erim2 35913 |
Copyright terms: Public domain | W3C validator |