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 2390 and spimfv 2235. (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 1914	. 2 ⊢ (¬ 𝜓 → ∀𝑥 ¬ 𝜓)
2		spimvw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimw 1975	1 ⊢ (∀𝑥𝜑 → 𝜓)

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972

This theorem depends on definitions:  df-bi 206  df-ex 1784

This theorem is referenced by:  spvv  2001  cbvalivw  2011  alcomiw  2047  alcomiwOLD  2048  axc16i  2436  ax9ALT  2733  reu6  3656  disj  4378  el  5287  fvn0ssdmfun  6934  aev-o  36872  axc11next  41913  funressnvmo  44426