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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 3958 | . 2 ⊢ ((𝐴 = 𝐵 ∧ 𝐶 = 𝐷) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) | |
4 | 1, 2, 3 | syl2an 575 | 1 ⊢ ((𝜑 ∧ 𝜓) → (𝐴 ∩ 𝐶) = (𝐵 ∩ 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 = wceq 1630 ∩ cin 3720 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-v 3351 df-in 3728 |
This theorem is referenced by: funprg 6083 funtpg 6084 funcnvpr 6091 funcnvqp 6093 fvun1 6411 fndmin 6467 offval 7050 ofrfval 7051 offval3 7308 fpar 7431 wfrlem4 7569 wfrlem4OLD 7570 fisn 8488 ixxin 12396 vdwmc 15888 fvcosymgeq 18055 cssincl 20248 inmbl 23529 iundisj2 23536 itg1addlem3 23684 fh1 28811 iundisj2f 29735 iundisj2fi 29890 br1cosscnvxrn 34559 offval0 42817 |
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