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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 4020 | . 2 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐶)) |
3 | psseq12d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | psseq2d 4021 | . 2 ⊢ (𝜑 → (𝐵 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) |
5 | 2, 4 | bitrd 282 | 1 ⊢ (𝜑 → (𝐴 ⊊ 𝐶 ↔ 𝐵 ⊊ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1538 ⊊ wpss 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-in 3888 df-ss 3898 df-pss 3900 |
This theorem is referenced by: fin23lem32 9755 fin23lem34 9757 fin23lem35 9758 fin23lem41 9763 isf32lem5 9768 isf32lem6 9769 isf32lem11 9774 compssiso 9785 chnle 29297 |
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