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Theorem goeleq12bg 35295
Description: Two "Godel-set of membership" codes for two variables are equal iff the two corresponding variables are equal. (Contributed by AV, 8-Oct-2023.)
Assertion
Ref Expression
goeleq12bg (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))

Proof of Theorem goeleq12bg
StepHypRef Expression
1 goel 35293 . . 3 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (𝐼𝑔𝐽) = ⟨∅, ⟨𝐼, 𝐽⟩⟩)
2 goel 35293 . . 3 ((𝑀 ∈ ω ∧ 𝑁 ∈ ω) → (𝑀𝑔𝑁) = ⟨∅, ⟨𝑀, 𝑁⟩⟩)
31, 2eqeqan12rd 2748 . 2 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ ⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩))
4 0ex 5308 . . . 4 ∅ ∈ V
5 opex 5467 . . . 4 𝐼, 𝐽⟩ ∈ V
64, 5opth 5479 . . 3 (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
7 eqid 2733 . . . . 5 ∅ = ∅
87biantrur 530 . . . 4 (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩))
9 opthg 5480 . . . . 5 ((𝐼 ∈ ω ∧ 𝐽 ∈ ω) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
109adantl 481 . . . 4 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
118, 10bitr3id 285 . . 3 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((∅ = ∅ ∧ ⟨𝐼, 𝐽⟩ = ⟨𝑀, 𝑁⟩) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
126, 11bitrid 283 . 2 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → (⟨∅, ⟨𝐼, 𝐽⟩⟩ = ⟨∅, ⟨𝑀, 𝑁⟩⟩ ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
133, 12bitrd 279 1 (((𝑀 ∈ ω ∧ 𝑁 ∈ ω) ∧ (𝐼 ∈ ω ∧ 𝐽 ∈ ω)) → ((𝐼𝑔𝐽) = (𝑀𝑔𝑁) ↔ (𝐼 = 𝑀𝐽 = 𝑁)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1535  wcel 2104  c0 4339  cop 4636  (class class class)co 7425  ωcom 7880  𝑔cgoe 35279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-10 2137  ax-11 2153  ax-12 2173  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1087  df-tru 1538  df-fal 1548  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2536  df-eu 2565  df-clab 2711  df-cleq 2725  df-clel 2812  df-nfc 2888  df-ral 3058  df-rex 3067  df-rab 3433  df-v 3479  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4915  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-iota 6510  df-fun 6560  df-fv 6566  df-ov 7428  df-goel 35286
This theorem is referenced by:  satfv0  35304  satfv0fun  35317
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