Theorem spimevw 1985

Description: Existential introduction, using implicit substitution. This is to spimew 1971 what spimvw 1986 is to spimw 1970. Version of spimev 2390 and spimefv 2199 with an additional disjoint variable condition, using only Tarski's FOL axiom schemes. (Contributed by NM, 10-Jan-1993.) (Revised by BJ, 17-Mar-2020.)

Hypothesis

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

Assertion

Ref	Expression
spimevw	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem spimevw

Step	Hyp	Ref	Expression
1		ax-5 1910	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		spimevw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimew 1971	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1779

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967

This theorem depends on definitions:  df-bi 207  df-ex 1780

This theorem is referenced by:  dtruALT2  5309  zfpair  5360  axprlem3  5364  exneq  5379  fvn0ssdmfun  7008  onsupmaxb  43222  refimssco  43590  rlimdmafv  47171  rlimdmafv2  47252  elsprel  47469