Theorem bj-spimevw 32657

Description: Existential introduction, using implicit substitution. This is to spimeh 1925 what spimvw 1927 is to spimw 1926. (Contributed by BJ, 17-Mar-2020.)

Hypothesis

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

Assertion

Ref	Expression
bj-spimevw	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

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

Proof of Theorem bj-spimevw

Step	Hyp	Ref	Expression
1		ax-5 1839	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		bj-spimevw.1	. 2 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
3	1, 2	spimeh 1925	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∃wex 1704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888

This theorem depends on definitions:  df-bi 197  df-ex 1705

This theorem is referenced by: (None)