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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 4084 | . 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 205 = wceq 1533 ⊊ wpss 3945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-9 2108 ax-ext 2696 |
This theorem depends on definitions: df-bi 206 df-an 395 df-ex 1774 df-cleq 2717 df-ne 2930 df-ss 3961 df-pss 3964 |
This theorem is referenced by: psseq12d 4090 php3 9237 php3OLD 9249 inf3lem5 9657 infeq5i 9661 ackbij1lem15 10259 fin4en1 10334 chpsscon1 31386 chnle 31396 atcvatlem 32267 atcvati 32268 lsatcvat 38652 |
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