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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 4010 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 ⊆ 𝐴 ↔ 𝑅 ⊆ 𝐵)) | |
| 2 | cleq2lem.b | . 2 ⊢ (𝐴 = 𝐵 → (𝜑 ↔ 𝜓)) | |
| 3 | 1, 2 | anbi12d 632 | 1 ⊢ (𝐴 = 𝐵 → ((𝑅 ⊆ 𝐴 ∧ 𝜑) ↔ (𝑅 ⊆ 𝐵 ∧ 𝜓))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ⊆ wss 3951 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ss 3968 | 
| This theorem is referenced by: cbvcllem 43622 clublem 43623 rclexi 43628 rtrclex 43630 rtrclexi 43634 clrellem 43635 clcnvlem 43636 trcleq2lemRP 43643 dfrcl2 43687 brtrclfv2 43740 clsk1indlem1 44058 | 
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