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Theorem spim 2397
 Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2397 series of theorems requires that only one direction of the substitution hypothesis hold. Usage of this theorem is discouraged because it depends on ax-13 2382. Check out spimw 1973 for a version requiring less axioms. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-2018.) (New usage is discouraged.)
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 . 2 𝑥𝜓
2 ax6e 2393 . . 3 𝑥 𝑥 = 𝑦
3 spim.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1838 . 2 𝑥(𝜑𝜓)
51, 419.36i 2232 1 (∀𝑥𝜑𝜓)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536  Ⅎwnf 1785 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-12 2176  ax-13 2382 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-nf 1786 This theorem is referenced by:  spimv  2400  chvar  2405  cbv3  2407
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