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| Mirrors > Home > MPE Home > Th. List > psseq2i | 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 | 
|---|---|
| psseq2i | ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | psseq1i.1 | . 2 ⊢ 𝐴 = 𝐵 | |
| 2 | psseq2 4091 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) | |
| 3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ 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: psseq12i 4094 disjpss 4461 infeq5i 9676 | 
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