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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 4034 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑆 ⊆ 𝑅) | |
2 | sess1 5635 | . . 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 4033 | . . 3 ⊢ (𝑅 = 𝑆 → 𝑅 ⊆ 𝑆) | |
5 | sess1 5635 | . . 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 1533 ⊆ wss 3941 Se wse 5620 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1536 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ral 3054 df-rab 3425 df-v 3468 df-in 3948 df-ss 3958 df-br 5140 df-se 5623 |
This theorem is referenced by: oieq1 9504 |
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