![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > cleq2lem | Structured version Visualization version GIF version |
Description: Equality implies bijection. (Contributed by RP, 24-Jul-2020.) |
Ref | Expression |
---|---|
cleq2lem.b | ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) |
Ref | Expression |
---|---|
cleq2lem | ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq2 4035 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵)) | |
2 | cleq2lem.b | . 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 631 | 1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ⊆ wss 3976 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2711 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1778 df-cleq 2732 df-ss 3993 |
This theorem is referenced by: cbvcllem 43571 clublem 43572 rclexi 43577 rtrclex 43579 rtrclexi 43583 clrellem 43584 clcnvlem 43585 trcleq2lemRP 43592 dfrcl2 43636 brtrclfv2 43689 clsk1indlem1 44007 |
Copyright terms: Public domain | W3C validator |