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Mirrors > Home > MPE Home > Th. List > psseq12i | 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 |
psseq12i.2 | ⊢ 𝐶 = 𝐷 |
Ref | Expression |
---|---|
psseq12i | ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq1i.1 | . . 3 ⊢ 𝐴 = 𝐵 | |
2 | 1 | psseq1i 4068 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶) |
3 | psseq12i.2 | . . 3 ⊢ 𝐶 = 𝐷 | |
4 | 3 | psseq2i 4069 | . 2 ⊢ (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
5 | 2, 4 | bitri 277 | 1 ⊢ (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ⊊ wpss 3939 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ne 3019 df-in 3945 df-ss 3954 df-pss 3956 |
This theorem is referenced by: canthp1lem2 10077 symgvalstruct 18527 |
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