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Mirrors > Home > MPE Home > Th. List > psseq2d | 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) |
Ref | Expression |
---|---|
psseq2d | ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psseq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | psseq2 3981 | . 2 ⊢ (𝐴 = 𝐵 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝐶 ⊊ 𝐴 ↔ 𝐶 ⊊ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 = wceq 1520 ⊊ wpss 3855 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1775 ax-4 1789 ax-5 1886 ax-6 1945 ax-7 1990 ax-8 2081 ax-9 2089 ax-10 2110 ax-11 2124 ax-12 2139 ax-ext 2767 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1523 df-ex 1760 df-nf 1764 df-sb 2041 df-clab 2774 df-cleq 2786 df-clel 2861 df-ne 2983 df-in 3861 df-ss 3869 df-pss 3871 |
This theorem is referenced by: psseq12d 3987 php3 8540 inf3lem5 8930 infeq5i 8934 ackbij1lem15 9491 fin4en1 9566 chpsscon1 28960 chnle 28970 atcvatlem 29841 atcvati 29842 lsatcvat 35667 |
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