Theorem spimevw 1987

Description: Existential introduction, using implicit substitution. This is to spimew 1973 what spimvw 1988 is to spimw 1972. Version of spimev 2396 and spimefv 2206 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 1912	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		spimevw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimew 1973	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1781

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969

This theorem depends on definitions:  df-bi 207  df-ex 1782

This theorem is referenced by:  dtruALT2  5312  zfpair  5363  axprlem3  5367  exneq  5388  fvn0ssdmfun  7026  axnulregtco  36662  onsupmaxb  43667  refimssco  44034  rlimdmafv  47625  rlimdmafv2  47706  elsprel  47935