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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 3627 | . 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 384 = wceq 1483 ⊆ wss 3574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-in 3581 df-ss 3588 |
This theorem is referenced by: cbvcllem 37915 clublem 37917 rclexi 37922 rtrclex 37924 rtrclexi 37928 clrellem 37929 clcnvlem 37930 trcleq2lemRP 37937 dfrcl2 37966 brtrclfv2 38019 clsk1indlem1 38343 |
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