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Mirrors > Home > MPE Home > Th. List > cleq1lem | Structured version Visualization version GIF version |
Description: Equality implies bijection. (Contributed by RP, 9-May-2020.) |
Ref | Expression |
---|---|
cleq1lem | ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3995 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | 1 | anbi1d 631 | 1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ⊆ wss 3939 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-in 3946 df-ss 3955 |
This theorem is referenced by: cleq1 14346 trcleq12lem 14356 lcmfun 15992 coprmproddvds 16010 isslw 18736 neival 21713 nrmsep3 21966 xkococnlem 22270 ovolval 24077 ovnval2b 42841 |
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