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Theorem chvar 1688
 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 1674 . 2 (∀𝑥𝜑𝜓)
5 chvar.3 . 2 𝜑
64, 5mpg 1386 1 𝜓
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 104  Ⅎwnf 1395 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1382  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-4 1446  ax-i9 1469  ax-ial 1473 This theorem depends on definitions:  df-bi 116  df-nf 1396 This theorem is referenced by:  csbhypf  2969  opelopabsb  4098  findes  4433  fvmptssdm  5402  dfoprab4f  5979  dom2lem  6545  uzind4s  9141  fsumsplitf  10865
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