Theorem abbiOLD 2954

Description: Obsolete proof of abbi 2887 as of 7-Jan-2024. (Contributed by NM, 25-Nov-2013.) (Revised by Mario Carneiro, 11-Aug-2016.) (Proof shortened by Wolf Lammen, 16-Nov-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem abbiOLD

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hbab1 2806	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜑} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜑})
2		hbab1 2806	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ 𝜓} → ∀𝑥 𝑦 ∈ {𝑥 ∣ 𝜓})
3	1, 2	cleqh 2935	. 2 ⊢ ({𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓} ↔ ∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}))
4		abid 2802	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑)
5		abid 2802	. . . 4 ⊢ (𝑥 ∈ {𝑥 ∣ 𝜓} ↔ 𝜓)
6	4, 5	bibi12i 342	. . 3 ⊢ ((𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ (𝜑 ↔ 𝜓))
7	6	albii 1819	. 2 ⊢ (∀𝑥(𝑥 ∈ {𝑥 ∣ 𝜑} ↔ 𝑥 ∈ {𝑥 ∣ 𝜓}) ↔ ∀𝑥(𝜑 ↔ 𝜓))
8	3, 7	bitr2i 278	1 ⊢ (∀𝑥(𝜑 ↔ 𝜓) ↔ {𝑥 ∣ 𝜑} = {𝑥 ∣ 𝜓})

Syntax hints:   ↔ wb 208  ∀wal 1534   = wceq 1536   ∈ wcel 2113  {cab 2798

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892

This theorem is referenced by: (None)