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| Mirrors > Home > MPE Home > Th. List > psseq1d | Structured version Visualization version GIF version | ||
| Description: An equality deduction for the proper subclass relationship. (Contributed by NM, 9-Jun-2004.) |
| Ref | Expression |
|---|---|
| psseq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
| Ref | Expression |
|---|---|
| psseq1d | ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | psseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
| 2 | psseq1 4090 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
| 3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1540 ⊊ wpss 3952 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1780 df-cleq 2729 df-ne 2941 df-ss 3968 df-pss 3971 |
| This theorem is referenced by: psseq12d 4097 fin23lem32 10384 fin23lem35 10387 compssiso 10414 mrieqv2d 17682 mrissmrcd 17683 pgpfac1lem5 20099 islbs3 21157 chpsscon2 31524 |
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