Theorem 19.9dev 43019

Description: 19.9d 2242 in the case of an existential quantifier, avoiding the ax-10 2179 from nfex 2359 that would be used for the hypothesis of 19.9d 2242, at the cost of an additional DV condition on 𝑦, 𝜑. (Contributed by SN, 26-May-2024.)

Hypothesis

Ref	Expression
19.9dev.1	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
19.9dev	⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓))

Distinct variable group:   𝜑,𝑦

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

Proof of Theorem 19.9dev

Step	Hyp	Ref	Expression
1		excom 2200	. 2 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
2		19.9dev.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		19.9t 2243	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ 𝜓))
5	4	exbidv 1954	. 2 ⊢ (𝜑 → (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓))
6	1, 5	bitrid 286	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209  ∃wex 1812  Ⅎwnf 1816

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-11 2195  ax-12 2216

This proof depends on definitions:  df-bi 210  df-ex 1813  df-nf 1817

This theorem is used by: (None)