Theorem cleljustALT2 2383

Description: Alternate proof of cleljust 2123. Compared with cleljustALT 2382, it uses nfv 1915 followed by equsexv 2269 instead of ax-5 1911 followed by equsexhv 2300, so it uses the idiom Ⅎ𝑥𝜑 instead of 𝜑 → ∀𝑥𝜑 to express non-freeness. This style is generally preferred for later theorems. (Contributed by NM, 28-Jan-2004.) (Revised by Mario Carneiro, 21-Dec-2016.) (Revised by BJ, 29-Dec-2020.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
cleljustALT2	⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

Distinct variable groups:   𝑥,𝑧   𝑦,𝑧

Proof of Theorem cleljustALT2

Step	Hyp	Ref	Expression
1		nfv 1915	. . 3 ⊢ Ⅎ𝑧 𝑥 ∈ 𝑦
2		elequ1 2121	. . 3 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ 𝑦 ↔ 𝑥 ∈ 𝑦))
3	1, 2	equsexv 2269	. 2 ⊢ (∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦) ↔ 𝑥 ∈ 𝑦)
4	3	bicomi 226	1 ⊢ (𝑥 ∈ 𝑦 ↔ ∃𝑧(𝑧 = 𝑥 ∧ 𝑧 ∈ 𝑦))

Syntax hints:   ↔ wb 208   ∧ wa 398  ∃wex 1780

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785

This theorem is referenced by: (None)