![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > eqopab2b | Structured version Visualization version GIF version |
Description: Equivalence of ordered pair abstraction equality and biconditional. Usage of this theorem is discouraged because it depends on ax-13 2365. Use the weaker eqopab2bw 5541 when possible. (Contributed by Mario Carneiro, 4-Jan-2017.) (New usage is discouraged.) |
Ref | Expression |
---|---|
eqopab2b | ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssopab2b 5542 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 → 𝜓)) | |
2 | ssopab2b 5542 | . . 3 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑} ↔ ∀𝑥∀𝑦(𝜓 → 𝜑)) | |
3 | 1, 2 | anbi12i 626 | . 2 ⊢ (({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑}) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) |
4 | eqss 3992 | . 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ∧ {⟨𝑥, 𝑦⟩ ∣ 𝜓} ⊆ {⟨𝑥, 𝑦⟩ ∣ 𝜑})) | |
5 | 2albiim 1885 | . 2 ⊢ (∀𝑥∀𝑦(𝜑 ↔ 𝜓) ↔ (∀𝑥∀𝑦(𝜑 → 𝜓) ∧ ∀𝑥∀𝑦(𝜓 → 𝜑))) | |
6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ ({⟨𝑥, 𝑦⟩ ∣ 𝜑} = {⟨𝑥, 𝑦⟩ ∣ 𝜓} ↔ ∀𝑥∀𝑦(𝜑 ↔ 𝜓)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∀wal 1531 = wceq 1533 ⊆ wss 3943 {copab 5203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-13 2365 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ral 3056 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-opab 5204 |
This theorem is referenced by: opabbi 37545 |
Copyright terms: Public domain | W3C validator |