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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 4068 | . 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 208 = wceq 1536 ⊊ wpss 3940 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-ne 3020 df-in 3946 df-ss 3955 df-pss 3957 |
This theorem is referenced by: psseq12i 4071 disjpss 4413 infeq5i 9102 |
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