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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjeqd | Structured version Visualization version GIF version |
Description: Equality theorem for disjoint elementhood, deduction version. (Contributed by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
eldisjeqd.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
Ref | Expression |
---|---|
eldisjeqd | ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eldisjeqd.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | eldisjeq 37416 | . 2 ⊢ (𝐴 = 𝐵 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → ( ElDisj 𝐴 ↔ ElDisj 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1541 ElDisj weldisj 36884 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-coss 37086 df-cnvrefrel 37202 df-funALTV 37357 df-disjALTV 37380 df-eldisj 37382 |
This theorem is referenced by: (None) |
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