Theorem 19.9dev 40106

Description: 19.9d 2199 in the case of an existential quantifier, avoiding the ax-10 2139 from nfex 2322 that would be used for the hypothesis of 19.9d 2199, 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 2164	. 2 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
2		19.9dev.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		19.9t 2200	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ 𝜓))
5	4	exbidv 1925	. 2 ⊢ (𝜑 → (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓))
6	1, 5	syl5bb 282	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 205  ∃wex 1783  Ⅎwnf 1787

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-11 2156  ax-12 2173

This theorem depends on definitions:  df-bi 206  df-ex 1784  df-nf 1788

This theorem is referenced by: (None)