Theorem spimehOLD 1975

Description: Obsolete version of spimew 1974 as of 22-Oct-2023. (Contributed by NM, 7-Aug-1994.) (Proof shortened by Wolf Lammen, 10-Dec-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
spimew.1	⊢ (𝜑 → ∀𝑥𝜑)
spimew.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
spimehOLD	⊢ (𝜑 → ∃𝑥𝜓)

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem spimehOLD

Step	Hyp	Ref	Expression
1		spimew.1	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
2		ax6ev 1972	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
3		spimew.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
4	2, 3	eximii 1837	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
5	4	19.35i 1879	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
6	1, 5	syl 17	1 ⊢ (𝜑 → ∃𝑥𝜓)

Syntax hints:   → wi 4  ∀wal 1535  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-6 1970

This theorem depends on definitions:  df-bi 209  df-ex 1781

This theorem is referenced by: (None)