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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 4087 | . 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 205 = wceq 1539 ⊊ wpss 3948 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-ne 2939 df-v 3474 df-in 3954 df-ss 3964 df-pss 3966 |
This theorem is referenced by: psseq12i 4090 disjpss 4459 infeq5i 9633 |
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