Theorem spimvw 2019

Description: A weak form of specialization. Lemma 8 of [KalishMontague] p. 87. Uses only Tarski's FOL axiom schemes. For stronger forms using more axioms, see spimv 2424 and spimfv 2278. (Contributed by NM, 9-Apr-2017.)

Hypothesis

Ref	Expression
spimvw.1	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimvw	⊢ (∀𝑥𝜑 → 𝜓)

Distinct variable groups:   𝑥,𝑦   𝜓,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝜓(𝑦)

Proof of Theorem spimvw

Step	Hyp	Ref	Expression
1		ax-5 1943	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		spimvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimw 2003	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1568

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000

This proof depends on definitions:  df-bi 210  df-ex 1813

This theorem is used by:  spsv  2020  spvv  2021  cbvalivw  2040  alcomimw  2076  axc16i  2470  ax9ALT  2760  reu6  3691  exnelv  5278  elALT2  5342  el  5421  fvn0ssdmfun  7073  axtco1from2  37047  wl-dfcleq  38221  aev-o  39767  axc11next  45193  funressnvmo  47859