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Theorem spim 2242
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2242 series of theorems requires that only one direction of the substitution hypothesis hold. (Contributed by NM, 10-Jan-1993.) (Revised by Mario Carneiro, 3-Oct-2016.) (Proof shortened by Wolf Lammen, 18-Feb-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 . 2 𝑥𝜓
2 ax6e 2238 . . 3 𝑥 𝑥 = 𝑦
3 spim.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1754 . 2 𝑥(𝜑𝜓)
51, 419.36i 2086 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1473  wnf 1699
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-12 2034  ax-13 2234
This theorem depends on definitions:  df-bi 196  df-an 385  df-ex 1696  df-nf 1701
This theorem is referenced by:  spimvALT  2246  chvar  2250  cbv3  2253
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