New Foundations Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > NFE Home > Th. List > eqvinop | GIF version |
Description: A variable introduction law for ordered pairs. Analog of Lemma 15 of [Monk2] p. 109. (Contributed by NM, 28-May-1995.) |
Ref | Expression |
---|---|
eqvinop.1 | ⊢ B ∈ V |
eqvinop.2 | ⊢ C ∈ V |
Ref | Expression |
---|---|
eqvinop | ⊢ (A = 〈B, C〉 ↔ ∃x∃y(A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | opth 4602 | . . . . . . . 8 ⊢ (〈x, y〉 = 〈B, C〉 ↔ (x = B ∧ y = C)) | |
2 | ancom 437 | . . . . . . . 8 ⊢ ((x = B ∧ y = C) ↔ (y = C ∧ x = B)) | |
3 | 1, 2 | bitri 240 | . . . . . . 7 ⊢ (〈x, y〉 = 〈B, C〉 ↔ (y = C ∧ x = B)) |
4 | 3 | anbi2i 675 | . . . . . 6 ⊢ ((A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉) ↔ (A = 〈x, y〉 ∧ (y = C ∧ x = B))) |
5 | an13 774 | . . . . . 6 ⊢ ((A = 〈x, y〉 ∧ (y = C ∧ x = B)) ↔ (x = B ∧ (y = C ∧ A = 〈x, y〉))) | |
6 | 4, 5 | bitri 240 | . . . . 5 ⊢ ((A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉) ↔ (x = B ∧ (y = C ∧ A = 〈x, y〉))) |
7 | 6 | exbii 1582 | . . . 4 ⊢ (∃y(A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉) ↔ ∃y(x = B ∧ (y = C ∧ A = 〈x, y〉))) |
8 | 19.42v 1905 | . . . 4 ⊢ (∃y(x = B ∧ (y = C ∧ A = 〈x, y〉)) ↔ (x = B ∧ ∃y(y = C ∧ A = 〈x, y〉))) | |
9 | eqvinop.2 | . . . . . 6 ⊢ C ∈ V | |
10 | opeq2 4579 | . . . . . . 7 ⊢ (y = C → 〈x, y〉 = 〈x, C〉) | |
11 | 10 | eqeq2d 2364 | . . . . . 6 ⊢ (y = C → (A = 〈x, y〉 ↔ A = 〈x, C〉)) |
12 | 9, 11 | ceqsexv 2894 | . . . . 5 ⊢ (∃y(y = C ∧ A = 〈x, y〉) ↔ A = 〈x, C〉) |
13 | 12 | anbi2i 675 | . . . 4 ⊢ ((x = B ∧ ∃y(y = C ∧ A = 〈x, y〉)) ↔ (x = B ∧ A = 〈x, C〉)) |
14 | 7, 8, 13 | 3bitri 262 | . . 3 ⊢ (∃y(A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉) ↔ (x = B ∧ A = 〈x, C〉)) |
15 | 14 | exbii 1582 | . 2 ⊢ (∃x∃y(A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉) ↔ ∃x(x = B ∧ A = 〈x, C〉)) |
16 | eqvinop.1 | . . 3 ⊢ B ∈ V | |
17 | opeq1 4578 | . . . 4 ⊢ (x = B → 〈x, C〉 = 〈B, C〉) | |
18 | 17 | eqeq2d 2364 | . . 3 ⊢ (x = B → (A = 〈x, C〉 ↔ A = 〈B, C〉)) |
19 | 16, 18 | ceqsexv 2894 | . 2 ⊢ (∃x(x = B ∧ A = 〈x, C〉) ↔ A = 〈B, C〉) |
20 | 15, 19 | bitr2i 241 | 1 ⊢ (A = 〈B, C〉 ↔ ∃x∃y(A = 〈x, y〉 ∧ 〈x, y〉 = 〈B, C〉)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 176 ∧ wa 358 ∃wex 1541 = wceq 1642 ∈ wcel 1710 Vcvv 2859 〈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-13 1712 ax-14 1714 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-3or 935 df-3an 936 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-eu 2208 df-mo 2209 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-ne 2518 df-ral 2619 df-rex 2620 df-reu 2621 df-rmo 2622 df-rab 2623 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-pss 3261 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-iota 4339 df-0c 4377 df-addc 4378 df-nnc 4379 df-fin 4380 df-lefin 4440 df-ltfin 4441 df-ncfin 4442 df-tfin 4443 df-evenfin 4444 df-oddfin 4445 df-sfin 4446 df-spfin 4447 df-phi 4565 df-op 4566 df-proj1 4567 df-proj2 4568 |
This theorem is referenced by: copsexg 4607 ralxpf 4827 oprabid 5550 |
Copyright terms: Public domain | W3C validator |