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Theorem spim 1626
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 1626 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 1412 . 2 (𝜓 → ∀𝑥𝜓)
3 spim.2 . 2 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3spimh 1625 1 (∀𝑥𝜑𝜓)
Colors of variables: wff set class
Syntax hints:  wi 4  wal 1241  wnf 1349
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1336  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-4 1400  ax-i9 1423  ax-ial 1427
This theorem depends on definitions:  df-bi 110  df-nf 1350
This theorem is referenced by:  cbv3  1630  chvar  1640  spimv  1692  2spim  9906
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