Theorem spimvw 2000

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 2415 and spimfv 2268. (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 1924	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		spimvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimw 1984	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1552

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1809  ax-4 1823  ax-5 1924  ax-6 1981

This theorem depends on definitions:  df-bi 209  df-ex 1794

This theorem is referenced by:  spsv  2001  spvv  2002  cbvalivw  2021  alcomimw  2057  axc16i  2461  ax9ALT  2751  reu6  3683  exnelv  5257  elALT2  5320  el  5399  fvn0ssdmfun  7044  axtco1from2  36783  wl-dfcleq  37956  aev-o  39503  axc11next  44930  funressnvmo  47587