Theorem spimevw 1994

Description: Existential introduction, using implicit substitution. This is to spimew 1971 what spimvw 1995 is to spimw 1970. Version of spimev 2397 and spimefv 2198 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  5370  zfpair  5421  axprlem3  5425  exneq  5440  dtruOLD  5446  fvn0ssdmfun  7094  onsupmaxb  43251  refimssco  43620  rlimdmafv  47189  rlimdmafv2  47270  elsprel  47462