Theorem bj-spimedv 33056

Description: Version of spimed 2417 with a dv condition, which does not require ax-13 2408. (Contributed by BJ, 31-May-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-spimedv.1	⊢ (𝜒 → Ⅎ𝑥𝜑)
bj-spimedv.2	⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))

Assertion

Ref	Expression
bj-spimedv	⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Distinct variable group:   𝑥,𝑦

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

Proof of Theorem bj-spimedv

Step	Hyp	Ref	Expression
1		bj-spimedv.1	. . 3 ⊢ (𝜒 → Ⅎ𝑥𝜑)
2	1	nf5rd 2220	. 2 ⊢ (𝜒 → (𝜑 → ∀𝑥𝜑))
3		ax6ev 2059	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑦
4		bj-spimedv.2	. . . 4 ⊢ (𝑥 = 𝑦 → (𝜑 → 𝜓))
5	3, 4	eximii 1912	. . 3 ⊢ ∃𝑥(𝜑 → 𝜓)
6	5	19.35i 1958	. 2 ⊢ (∀𝑥𝜑 → ∃𝑥𝜓)
7	2, 6	syl6 35	1 ⊢ (𝜒 → (𝜑 → ∃𝑥𝜓))

Syntax hints:   → wi 4  ∀wal 1629  ∃wex 1852  Ⅎwnf 1856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-12 2203

This theorem depends on definitions:  df-bi 197  df-ex 1853  df-nf 1858

This theorem is referenced by:  bj-spimev  33057