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Theorem spim 1667
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1667 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 5-Aug-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof rewritten by Jim Kingdon, 10-Jun-2018.)
Hypotheses
Ref Expression
spim.1 𝑥𝜓
spim.2 (𝑥 = 𝑦 → (𝜑𝜓))
Assertion
Ref Expression
spim (∀𝑥𝜑𝜓)

Proof of Theorem spim
StepHypRef Expression
1 spim.1 . . 3 𝑥𝜓
21nfri 1453 . 2 (𝜓 → ∀𝑥𝜓)
3 spim.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimh 1666 1 (∀𝑥𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1283  wnf 1390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1377  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-4 1441  ax-i9 1464  ax-ial 1468
This theorem depends on definitions:  df-bi 115  df-nf 1391
This theorem is referenced by:  cbv3  1671  chvar  1681  spimv  1733  2spim  10728
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