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Theorem opelopabga 3963
 Description: The law of concretion. Theorem 9.5 of [Quine] p. 61. (Contributed by Mario Carneiro, 19-Dec-2013.)
Hypothesis
Ref Expression
opelopabga.1 ((x = A y = B) → (φψ))
Assertion
Ref Expression
opelopabga ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ ψ))
Distinct variable groups:   x,y,A   x,B,y   ψ,x,y
Allowed substitution hints:   φ(x,y)   𝑉(x,y)   𝑊(x,y)

Proof of Theorem opelopabga
StepHypRef Expression
1 elopab 3958 . 2 (⟨A, B {⟨x, y⟩ ∣ φ} ↔ xy(⟨A, B⟩ = ⟨x, y φ))
2 opelopabga.1 . . 3 ((x = A y = B) → (φψ))
32copsex2g 3946 . 2 ((A 𝑉 B 𝑊) → (xy(⟨A, B⟩ = ⟨x, y φ) ↔ ψ))
41, 3syl5bb 181 1 ((A 𝑉 B 𝑊) → (⟨A, B {⟨x, y⟩ ∣ φ} ↔ ψ))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1223  ∃wex 1354   ∈ wcel 1366  ⟨cop 3342  {copab 3780 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-14 1378  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995  ax-sep 3838  ax-pow 3890  ax-pr 3907 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-eu 1876  df-mo 1877  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-un 2890  df-in 2892  df-ss 2899  df-pw 3325  df-sn 3345  df-pr 3346  df-op 3348  df-opab 3782 This theorem is referenced by:  brabga  3964  opelopab2a  3965  opelopaba  3966  opelopabg  3968  isprmpt2  5768
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