Theorem abeq2f 2792

Description: Equality of a class variable and a class abstraction. In this version, the fact that 𝑥 is a non-free variable in 𝐴 is explicitly stated as a hypothesis. (Contributed by Thierry Arnoux, 11-May-2017.)

Hypothesis

Ref	Expression
abeq2f.0	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
abeq2f	⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Proof of Theorem abeq2f

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		abeq2f.0	. . . 4 ⊢ Ⅎ𝑥𝐴
2	1	nfcrii 2757	. . 3 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
3		hbab1 2611	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
4	2, 3	cleqh 2724	. 2 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}))
5		abid 2610	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
6	5	bibi2i 327	. . 3 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ (𝑥 ∈ 𝐴 ↔ 𝜑))
7	6	albii 1747	. 2 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑥 ∣ 𝜑}) ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))
8	4, 7	bitri 264	1 ⊢ (𝐴 = {𝑥 ∣ 𝜑} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝜑))

Syntax hints:   ↔ wb 196  ∀wal 1481   = wceq 1483   ∈ wcel 1990  {cab 2608  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  rabid2f  3119  mptfnf  6015