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Mirrors > Home > MPE Home > Th. List > psseq1i | 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 |
Ref | Expression |
---|---|
psseq1i | ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
2 | psseq1 4034 | . 2 ⊢ (𝐴 = 𝐵 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1540 ⊊ wpss 3899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-ext 2707 |
This theorem depends on definitions: df-bi 206 df-an 397 df-tru 1543 df-ex 1781 df-sb 2067 df-clab 2714 df-cleq 2728 df-clel 2814 df-ne 2941 df-v 3443 df-in 3905 df-ss 3915 df-pss 3917 |
This theorem is referenced by: psseq12i 4038 |
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