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Theorem spim 1718
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1718 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 1499 . 2 (𝜓 → ∀𝑥𝜓)
3 spim.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimh 1717 1 (∀𝑥𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1333  wnf 1440
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 1427  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-4 1490  ax-i9 1510  ax-ial 1514
This theorem depends on definitions:  df-bi 116  df-nf 1441
This theorem is referenced by:  cbv3  1722  chvar  1737  spimv  1791  2spim  13327
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