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Theorem cdeqab1 3714
 Description: Distribute conditional equality over abstraction. Usage of this theorem is discouraged because it depends on ax-13 2382. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.)
Hypothesis
Ref Expression
cdeqnot.1 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
cdeqab1 CondEq(𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜓})
Distinct variable groups:   𝜓,𝑥   𝜑,𝑦
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑦)

Proof of Theorem cdeqab1
StepHypRef Expression
1 nfv 1915 . . 3 𝑦𝜑
2 nfv 1915 . . 3 𝑥𝜓
3 cdeqnot.1 . . . 4 CondEq(𝑥 = 𝑦 → (𝜑𝜓))
43cdeqri 3708 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
51, 2, 4cbvab 2872 . 2 {𝑥𝜑} = {𝑦𝜓}
65cdeqth 3709 1 CondEq(𝑥 = 𝑦 → {𝑥𝜑} = {𝑦𝜓})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   = wceq 1538  {cab 2779  CondEqwcdeq 3705 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-13 2382  ax-ext 2773 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2780  df-cleq 2794  df-cdeq 3706 This theorem is referenced by: (None)
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