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Mirrors > Home > MPE Home > Th. List > tpeq123d | Structured version Visualization version GIF version |
Description: Equality theorem for unordered triples. (Contributed by NM, 22-Jun-2014.) |
Ref | Expression |
---|---|
tpeq1d.1 | ⊢ (𝜑 → 𝐴 = 𝐵) |
tpeq123d.2 | ⊢ (𝜑 → 𝐶 = 𝐷) |
tpeq123d.3 | ⊢ (𝜑 → 𝐸 = 𝐹) |
Ref | Expression |
---|---|
tpeq123d | ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | tpeq1d.1 | . . 3 ⊢ (𝜑 → 𝐴 = 𝐵) | |
2 | 1 | tpeq1d 4684 | . 2 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐶, 𝐸}) |
3 | tpeq123d.2 | . . 3 ⊢ (𝜑 → 𝐶 = 𝐷) | |
4 | 3 | tpeq2d 4685 | . 2 ⊢ (𝜑 → {𝐵, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐸}) |
5 | tpeq123d.3 | . . 3 ⊢ (𝜑 → 𝐸 = 𝐹) | |
6 | 5 | tpeq3d 4686 | . 2 ⊢ (𝜑 → {𝐵, 𝐷, 𝐸} = {𝐵, 𝐷, 𝐹}) |
7 | 2, 4, 6 | 3eqtrd 2863 | 1 ⊢ (𝜑 → {𝐴, 𝐶, 𝐸} = {𝐵, 𝐷, 𝐹}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 {ctp 4574 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-v 3499 df-un 3944 df-sn 4571 df-pr 4573 df-tp 4575 |
This theorem is referenced by: fz0tp 13011 fz0to4untppr 13013 fzo0to3tp 13126 fzo1to4tp 13128 prdsval 16731 imasval 16787 fucval 17231 fucpropd 17250 setcval 17340 catcval 17359 estrcval 17377 xpcval 17430 efmnd 18038 psrval 20145 om1val 23637 s3rn 30626 ldualset 36265 erngfset 37939 erngfset-rN 37947 dvafset 38144 dvaset 38145 dvhfset 38220 dvhset 38221 hlhilset 39074 rabren3dioph 39418 mendval 39789 nnsum4primesodd 43968 nnsum4primesoddALTV 43969 rngcvalALTV 44239 ringcvalALTV 44285 |
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