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Theorem chvar 1715
Description: Implicit substitution of 𝑦 for 𝑥 into a theorem. (Contributed by Raph Levien, 9-Jul-2003.) (Revised by Mario Carneiro, 3-Oct-2016.)
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 143 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3spim 1701 . 2 (∀𝑥𝜑𝜓)
5 chvar.3 . 2 𝜑
64, 5mpg 1412 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wnf 1421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-i9 1495  ax-ial 1499
This theorem depends on definitions:  df-bi 116  df-nf 1422
This theorem is referenced by:  csbhypf  3008  opelopabsb  4152  findes  4487  fvmptssdm  5473  dfoprab4f  6059  dom2lem  6634  uzind4s  9353  fsumsplitf  11145
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