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Theorem spim 2415
Description: Specialization, using implicit substitution. Compare Lemma 14 of [Tarski] p. 70. The spim 2415 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 2411 . . 3 𝑥 𝑥 = 𝑦
3 spim.2 . . 3 (𝑥 = 𝑦 → (𝜑𝜓))
42, 3eximii 1911 . 2 𝑥(𝜑𝜓)
51, 419.36i 2254 1 (∀𝑥𝜑𝜓)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1628  wnf 1855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-12 2202  ax-13 2407
This theorem depends on definitions:  df-bi 197  df-an 383  df-ex 1852  df-nf 1857
This theorem is referenced by:  spimv  2418  chvar  2423  cbv3  2425  setrec2fun  42957
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