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Mirrors > Home > MPE Home > Th. List > psseq12d | 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) |
psseq12d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
psseq12d | ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | psseq1d 3924 | . 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
3 | psseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | psseq2d 3925 | . 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) |
5 | 2, 4 | bitrd 271 | 1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1658 ⊊ wpss 3798 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-ext 2802 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-clab 2811 df-cleq 2817 df-clel 2820 df-ne 2999 df-in 3804 df-ss 3811 df-pss 3813 |
This theorem is referenced by: fin23lem32 9480 fin23lem34 9482 fin23lem35 9483 fin23lem41 9488 isf32lem5 9493 isf32lem6 9494 isf32lem11 9499 compssiso 9510 canthp1lem2 9789 chnle 28927 |
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