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Mirrors > Home > MPE Home > Th. List > seeq2 | Structured version Visualization version GIF version |
Description: Equality theorem for the set-like predicate. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
seeq2 | ⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqimss2 4068 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐵 ⊆ 𝐴) | |
2 | sess2 5666 | . . 3 ⊢ (𝐵 ⊆ 𝐴 → (𝑅 Se 𝐴 → 𝑅 Se 𝐵)) | |
3 | 1, 2 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 → 𝑅 Se 𝐵)) |
4 | eqimss 4067 | . . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ⊆ 𝐵) | |
5 | sess2 5666 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐵 → 𝑅 Se 𝐴)) |
7 | 3, 6 | impbid 212 | 1 ⊢ (𝐴 = 𝐵 → (𝑅 Se 𝐴 ↔ 𝑅 Se 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 = wceq 1537 ⊆ wss 3976 Se wse 5650 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-sep 5317 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-ex 1778 df-nf 1782 df-sb 2065 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ral 3068 df-rab 3444 df-v 3490 df-in 3983 df-ss 3993 df-se 5653 |
This theorem is referenced by: seeq12d 5672 oieq2 9582 |
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