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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 3660 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵)) | |
2 | cleq2lem.b | . 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
3 | 1, 2 | anbi12d 747 | 1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 = wceq 1523 ⊆ wss 3607 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-clab 2638 df-cleq 2644 df-clel 2647 df-in 3614 df-ss 3621 |
This theorem is referenced by: cbvcllem 38232 clublem 38234 rclexi 38239 rtrclex 38241 rtrclexi 38245 clrellem 38246 clcnvlem 38247 trcleq2lemRP 38254 dfrcl2 38283 brtrclfv2 38336 clsk1indlem1 38660 |
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