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Mirrors > Home > MPE Home > Th. List > seeq1 | Structured version Visualization version GIF version |
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
seeq1 | ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 4039 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑆 ⊆ 𝑅) | |
2 | sess1 5646 | . . 3 ⊢ (𝑆 ⊆ 𝑅 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 → 𝑆 Se 𝐴)) |
4 | eqimss 4038 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑅 ⊆ 𝑆) | |
5 | sess1 5646 | . . 3 ⊢ (𝑅 ⊆ 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑅 = 𝑆 → (𝑆 Se 𝐴 → 𝑅 Se 𝐴)) |
7 | 3, 6 | impbid 211 | 1 ⊢ (𝑅 = 𝑆 → (𝑅 Se 𝐴 ↔ 𝑆 Se 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ⊆ wss 3947 Se wse 5631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-tru 1537 df-ex 1775 df-nf 1779 df-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rab 3430 df-v 3473 df-in 3954 df-ss 3964 df-br 5149 df-se 5634 |
This theorem is referenced by: oieq1 9535 |
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