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Theorem rabeqbidva 2855
 Description: Equality of restricted class abstractions. (Contributed by Mario Carneiro, 26-Jan-2017.)
Hypotheses
Ref Expression
rabeqbidva.1 (φA = B)
rabeqbidva.2 ((φ x A) → (ψχ))
Assertion
Ref Expression
rabeqbidva (φ → {x A ψ} = {x B χ})
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hints:   ψ(x)   χ(x)

Proof of Theorem rabeqbidva
StepHypRef Expression
1 rabeqbidva.2 . . 3 ((φ x A) → (ψχ))
21rabbidva 2850 . 2 (φ → {x A ψ} = {x A χ})
3 rabeqbidva.1 . . 3 (φA = B)
4 rabeq 2853 . . 3 (A = B → {x A χ} = {x B χ})
53, 4syl 15 . 2 (φ → {x A χ} = {x B χ})
62, 5eqtrd 2385 1 (φ → {x A ψ} = {x B χ})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {crab 2618 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 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  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-ral 2619  df-rab 2623 This theorem is referenced by: (None)
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