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Mirrors > Home > MPE Home > Th. List > raleleq | Structured version Visualization version GIF version |
Description: All elements of a class are elements of a class equal to this class. (Contributed by AV, 30-Oct-2020.) |
Ref | Expression |
---|---|
raleleq | ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq2 2903 | . . 3 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 ↔ 𝑥 ∈ 𝐵)) | |
2 | 1 | biimpd 231 | . 2 ⊢ (𝐴 = 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) |
3 | 2 | ralrimiv 3183 | 1 ⊢ (𝐴 = 𝐵 → ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ∀wral 3140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-ex 1781 df-cleq 2816 df-clel 2895 df-ral 3145 |
This theorem is referenced by: uvtxnbgrb 27185 |
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