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Mirrors > Home > MPE Home > Th. List > psseq12i | Structured version Visualization version GIF version |
Description: An equality inference for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
Ref | Expression |
---|---|
psseq1i.1 | ⊢ 𝐴 = 𝐵 |
psseq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
psseq12i | ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | psseq1i 4084 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) |
3 | psseq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | psseq2i 4085 | . 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
5 | 2, 4 | bitri 275 | 1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ⊊ wpss 3944 |
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-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-v 3470 df-in 3950 df-ss 3960 df-pss 3962 |
This theorem is referenced by: canthp1lem2 10650 symgvalstruct 19316 symgvalstructOLD 19317 |
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