Theorem abbi 2766

Description: Equivalent wff's correspond to equal class abstractions. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.)

Assertion

Ref	Expression
abbi	⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Proof of Theorem abbi

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbab1 2640	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
2		hbab1 2640	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜓})
3	1, 2	cleqh 2753	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
4		abid 2639	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5		abid 2639	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
6	4, 5	bibi12i 328	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 ↔ 𝜓))
7	6	albii 1787	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 ↔ 𝜓))
8	3, 7	bitr2i 265	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Syntax hints:   ↔ wb 196  ∀wal 1521   = wceq 1523   ∈ wcel 2030  {cab 2637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647

This theorem is referenced by:  abbii  2768  abbid  2769  nabbi  2925  rabbi  3150  sbcbi2  3517  rabeqsn  4246  iuneq12df  4576  dfiota2  5890  iotabi  5898  uniabio  5899  iotanul  5904  karden  8796  iuneq12daf  29499  bj-cleq  33074  abeq12  34094  elnev  38956  csbingVD  39434  csbsngVD  39443  csbxpgVD  39444  csbrngVD  39446  csbunigVD  39448  csbfv12gALTVD  39449