Theorem spimvw 2020

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 2422 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 1944	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		spimvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimw 2004	1 ⊢ (∀𝑥𝜑 → 𝜓)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4  ∀wal 1567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001

This proof depends on definitions:  df-bi 210  df-ex 1812

This theorem is used by:  spsv  2021  spvv  2022  cbvalivw  2041  alcomimw  2077  axc16i  2468  ax9ALT  2758  reu6  3687  exnelv  5270  elALT2  5335  el  5410  fvn0ssdmfun  7085  reu6f  35765  axtco1from2  37396  wl-dfcleq  38583  aev-o  40134  axc11next  45536  funressnvmo  48260