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Theorem chvar 2405
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. Usage of this theorem is discouraged because it depends on ax-13 2382. Use the weaker chvarfv 2241 if possible. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.) (New usage is discouraged.)
Hypotheses
Ref Expression
chvar.1 𝑥𝜓
chvar.2 (𝑥 = 𝑦 → (𝜑𝜓))
chvar.3 𝜑
Assertion
Ref Expression
chvar 𝜓

Proof of Theorem chvar
StepHypRef Expression
1 chvar.1 . . 3 𝑥𝜓
2 chvar.2 . . . 4 (𝑥 = 𝑦 → (𝜑𝜓))
32biimpd 232 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3spim 2397 . 2 (∀𝑥𝜑𝜓)
5 chvar.3 . 2 𝜑
64, 5mpg 1799 1 𝜓
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wnf 1785
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-12 2176  ax-13 2382
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786
This theorem is referenced by:  chvarv  2406  zfcndrep  10029
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