Theorem abrexex2 7979

Description: Existence of an existentially restricted class abstraction. 𝜑 normally has free-variable parameters 𝑥 and 𝑦. See also abrexex 7972. (Contributed by NM, 12-Sep-2004.)

Hypotheses

Ref	Expression
abrexex2.1	⊢ 𝐴 ∈ V
abrexex2.2	⊢ {𝑦 ∣ 𝜑} ∈ V

Assertion

Ref	Expression
abrexex2	⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

Distinct variable group:   𝑥,𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥, 𝑦)

Proof of Theorem abrexex2

Step	Hyp	Ref	Expression
1		abrexex2.1	. 2 ⊢ 𝐴 ∈ V
2		abrexex2.2	. . 3 ⊢ {𝑦 ∣ 𝜑} ∈ V
3	2	rgenw 3081	. 2 ⊢ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V
4		abrexex2g 7974	. 2 ⊢ ((𝐴 ∈ V ∧ ∀𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ∈ V) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V)
5	1, 3, 4	mp2an 705	1 ⊢ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝜑} ∈ V

This proof depends on syntax axioms:   ∈ wcel 2145  {cab 2739  ∀wral 3077  ∃wrex 3087  Vcvv 3451

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-un 7749

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ral 3078  df-rex 3088  df-v 3453  df-ss 3916  df-uni 4868  df-iun 4953

This theorem is used by:  abexssex  7980  abexex  7981  oprabrexex2  7988  ab2rexex  7989  ab2rexex2  7990