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Mirrors > Home > NFE Home > Th. List > opelopabt | GIF version |
Description: Closed theorem form of opelopab 4708. (Contributed by NM, 19-Feb-2013.) |
Ref | Expression |
---|---|
opelopabt | ⊢ ((∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ)) ∧ (A ∈ V ∧ B ∈ W)) → (〈A, B〉 ∈ {〈x, y〉 ∣ φ} ↔ χ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elopab 4696 | . 2 ⊢ (〈A, B〉 ∈ {〈x, y〉 ∣ φ} ↔ ∃x∃y(〈A, B〉 = 〈x, y〉 ∧ φ)) | |
2 | 19.26-2 1594 | . . . . 5 ⊢ (∀x∀y((x = A → (φ ↔ ψ)) ∧ (y = B → (ψ ↔ χ))) ↔ (∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ)))) | |
3 | prth 554 | . . . . . . 7 ⊢ (((x = A → (φ ↔ ψ)) ∧ (y = B → (ψ ↔ χ))) → ((x = A ∧ y = B) → ((φ ↔ ψ) ∧ (ψ ↔ χ)))) | |
4 | bitr 689 | . . . . . . 7 ⊢ (((φ ↔ ψ) ∧ (ψ ↔ χ)) → (φ ↔ χ)) | |
5 | 3, 4 | syl6 29 | . . . . . 6 ⊢ (((x = A → (φ ↔ ψ)) ∧ (y = B → (ψ ↔ χ))) → ((x = A ∧ y = B) → (φ ↔ χ))) |
6 | 5 | 2alimi 1560 | . . . . 5 ⊢ (∀x∀y((x = A → (φ ↔ ψ)) ∧ (y = B → (ψ ↔ χ))) → ∀x∀y((x = A ∧ y = B) → (φ ↔ χ))) |
7 | 2, 6 | sylbir 204 | . . . 4 ⊢ ((∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ))) → ∀x∀y((x = A ∧ y = B) → (φ ↔ χ))) |
8 | copsex2t 4608 | . . . 4 ⊢ ((∀x∀y((x = A ∧ y = B) → (φ ↔ χ)) ∧ (A ∈ V ∧ B ∈ W)) → (∃x∃y(〈A, B〉 = 〈x, y〉 ∧ φ) ↔ χ)) | |
9 | 7, 8 | sylan 457 | . . 3 ⊢ (((∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ))) ∧ (A ∈ V ∧ B ∈ W)) → (∃x∃y(〈A, B〉 = 〈x, y〉 ∧ φ) ↔ χ)) |
10 | 9 | 3impa 1146 | . 2 ⊢ ((∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ)) ∧ (A ∈ V ∧ B ∈ W)) → (∃x∃y(〈A, B〉 = 〈x, y〉 ∧ φ) ↔ χ)) |
11 | 1, 10 | syl5bb 248 | 1 ⊢ ((∀x∀y(x = A → (φ ↔ ψ)) ∧ ∀x∀y(y = B → (ψ ↔ χ)) ∧ (A ∈ V ∧ B ∈ W)) → (〈A, B〉 ∈ {〈x, y〉 ∣ φ} ↔ χ)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 176 ∧ wa 358 ∧ w3a 934 ∀wal 1540 ∃wex 1541 = wceq 1642 ∈ wcel 1710 〈cop 4561 {copab 4622 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 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 df-opab 4623 |
This theorem is referenced by: fvopab4t 5385 |
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