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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 4100 | . 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 1537 ⊊ wpss 3964 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-ex 1777 df-cleq 2727 df-ne 2939 df-ss 3980 df-pss 3983 |
This theorem is referenced by: psseq12d 4107 fin23lem32 10382 fin23lem35 10385 compssiso 10412 mrieqv2d 17684 mrissmrcd 17685 pgpfac1lem5 20114 islbs3 21175 chpsscon2 31534 |
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