Theorem elintabg 41071

Description: Two ways of saying a set is an element of the intersection of a class. (Contributed by RP, 13-Aug-2020.)

Assertion

Ref	Expression
elintabg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥)   𝑉(𝑥)

Proof of Theorem elintabg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintg 4884	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦))
2		eleq2w 2822	. . 3 ⊢ (𝑦 = 𝑥 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑥))
3	2	ralab2 3627	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ 𝜑}𝐴 ∈ 𝑦 ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥))
4	1, 3	bitrdi 286	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝜑 → 𝐴 ∈ 𝑥)))

Syntax hints:   → wi 4   ↔ wb 205  ∀wal 1537   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∩ cint 4876

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-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-int 4877

This theorem is referenced by:  elinintab  41072