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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 4134 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | syl2an 598 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∩ cin 3880 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-tru 1541 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-rab 3115 df-in 3888 |
This theorem is referenced by: funprg 6378 funtpg 6379 funcnvpr 6386 funcnvqp 6388 fvun1 6729 fndmin 6792 offval 7396 ofrfval 7397 offval3 7665 fpar 7794 offsplitfpar 7798 wfrlem4 7941 fisn 8875 ixxin 12743 vdwmc 16304 fvcosymgeq 18549 cssincl 20377 inmbl 24146 iundisj2 24153 itg1addlem3 24302 fh1 29401 iundisj2f 30353 iundisj2fi 30546 satffunlem1lem1 32762 satffunlem2lem1 32764 br1cosscnvxrn 35874 |
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