Theorem 19.9dev 43363

Description: 19.9d 2239 in the case of an existential quantifier, avoiding the ax-10 2177 from nfex 2354 that would be used for the hypothesis of 19.9d 2239, 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 2198	. 2 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
2		19.9dev.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		19.9t 2240	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ 𝜓))
5	4	exbidv 1953	. 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 1811  Ⅎwnf 1815

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-11 2193  ax-12 2212

This proof depends on definitions:  df-bi 210  df-ex 1812  df-nf 1816

This theorem is used by: (None)