Theorem 19.9dev 42702

Description: 19.9d 2215 in the case of an existential quantifier, avoiding the ax-10 2152 from nfex 2333 that would be used for the hypothesis of 19.9d 2215, 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 2173	. 2 ⊢ (∃𝑥∃𝑦𝜓 ↔ ∃𝑦∃𝑥𝜓)
2		19.9dev.1	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3		19.9t 2216	. . . 4 ⊢ (Ⅎ𝑥𝜓 → (∃𝑥𝜓 ↔ 𝜓))
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → (∃𝑥𝜓 ↔ 𝜓))
5	4	exbidv 1928	. 2 ⊢ (𝜑 → (∃𝑦∃𝑥𝜓 ↔ ∃𝑦𝜓))
6	1, 5	bitrid 284	1 ⊢ (𝜑 → (∃𝑥∃𝑦𝜓 ↔ ∃𝑦𝜓))

Syntax hints:   → wi 4   ↔ wb 207  ∃wex 1786  Ⅎwnf 1790

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-11 2168  ax-12 2189

This theorem depends on definitions:  df-bi 208  df-ex 1787  df-nf 1791

This theorem is referenced by: (None)