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Mirrors > Home > MPE Home > Th. List > ineqan12d | Structured version Visualization version GIF version |
Description: Equality deduction for intersection of two classes. (Contributed by NM, 7-Feb-2007.) |
Ref | Expression |
---|---|
ineq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
ineqan12d.2 | ⊢ (𝜓 → 𝐶 = 𝐷) |
Ref | Expression |
---|---|
ineqan12d | ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ineq1d.1 | . 2 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | ineqan12d.2 | . 2 ⊢ (𝜓 → 𝐶 = 𝐷) | |
3 | ineq12 4222 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | syl2an 596 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1536 ∩ cin 3961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-rab 3433 df-in 3969 |
This theorem is referenced by: funprg 6621 funtpg 6622 funcnvpr 6629 funcnvqp 6631 fvun1 6999 fndmin 7064 ofrfvalg 7704 offval 7705 offval3 8005 fpar 8139 offsplitfpar 8142 wfrlem4OLD 8350 fisn 9464 ixxin 13400 vdwmc 17011 fvcosymgeq 19461 cssincl 21723 inmbl 25590 iundisj2 25597 itg1addlem3 25746 fh1 31646 iundisj2f 32609 of0r 32694 iundisj2fi 32804 satffunlem1lem1 35386 satffunlem2lem1 35388 disjeccnvep 38265 disjecxrn 38370 br1cosscnvxrn 38455 |
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