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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 3903 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊆ 𝐶 ↔ 𝐵 ⊆ 𝐶)) | |
2 | 1 | anbi1d 633 | 1 ⊢ (𝐴 = 𝐵 → ((𝐴 ⊆ 𝐶 ∧ 𝜑) ↔ (𝐵 ⊆ 𝐶 ∧ 𝜑))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1542 ⊆ wss 3844 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1545 df-ex 1787 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-v 3400 df-in 3851 df-ss 3861 |
This theorem is referenced by: cleq1 14433 trcleq12lem 14443 lcmfun 16087 coprmproddvds 16105 isslw 18852 neival 21854 nrmsep3 22107 xkococnlem 22411 ovolval 24226 ovnval2b 43624 |
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