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Mirrors > Home > NFE Home > Th. List > opeq12d | GIF version |
Description: Equality deduction for ordered pairs. (The proof was shortened by Andrew Salmon, 29-Jun-2011.) (Contributed by SF, 16-Dec-2006.) (Revised by SF, 29-Jun-2011.) |
Ref | Expression |
---|---|
opeq1d.1 | ⊢ (φ → A = B) |
opeq12d.2 | ⊢ (φ → C = D) |
Ref | Expression |
---|---|
opeq12d | ⊢ (φ → 〈A, C〉 = 〈B, D〉) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opeq1d.1 | . . 3 ⊢ (φ → A = B) | |
2 | 1 | opeq1d 4584 | . 2 ⊢ (φ → 〈A, C〉 = 〈B, C〉) |
3 | opeq12d.2 | . . 3 ⊢ (φ → C = D) | |
4 | 3 | opeq2d 4585 | . 2 ⊢ (φ → 〈B, C〉 = 〈B, D〉) |
5 | 2, 4 | eqtrd 2385 | 1 ⊢ (φ → 〈A, C〉 = 〈B, D〉) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1642 〈cop 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 ax-nin 4078 ax-xp 4079 ax-cnv 4080 ax-1c 4081 ax-sset 4082 ax-si 4083 ax-ins2 4084 ax-ins3 4085 ax-typlower 4086 ax-sn 4087 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-v 2861 df-sbc 3047 df-nin 3211 df-compl 3212 df-in 3213 df-un 3214 df-dif 3215 df-symdif 3216 df-ss 3259 df-nul 3551 df-if 3663 df-pw 3724 df-sn 3741 df-pr 3742 df-uni 3892 df-int 3927 df-opk 4058 df-1c 4136 df-pw1 4137 df-uni1 4138 df-xpk 4185 df-cnvk 4186 df-ins2k 4187 df-ins3k 4188 df-imak 4189 df-cok 4190 df-p6 4191 df-sik 4192 df-ssetk 4193 df-imagek 4194 df-idk 4195 df-addc 4378 df-nnc 4379 df-phi 4565 df-op 4566 |
This theorem is referenced by: nfopd 4605 1st2nd2 5516 xpassen 6057 |
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