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Theorem chvar 1656
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 136 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
41, 3spim 1642 . 2 (∀𝑥𝜑𝜓)
5 chvar.3 . 2 𝜑
64, 5mpg 1356 1 𝜓
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wnf 1365
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-5 1352  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-4 1416  ax-i9 1439  ax-ial 1443
This theorem depends on definitions:  df-bi 114  df-nf 1366
This theorem is referenced by:  csbhypf  2913  opelopabsb  4025  findes  4354  fvmptssdm  5283  dfoprab4f  5847  dom2lem  6283  uzind4s  8629
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