Theorem cleq2ab 1179

Description: Equality of two class abstractions means their wff's are equivalent.

Assertion

Ref	Expression
cleq2ab	⊢ ({x∣φ} = {x∣ψ} ↔ ∀x(φ ↔ ψ))

Proof of Theorem cleq2ab

Step	Hyp	Ref	Expression
1		hbab1 1095	. . 3 ⊢ (y ∈ {x∣φ} → ∀x y ∈ {x∣φ})
2		hbab1 1095	. . 3 ⊢ (y ∈ {x∣ψ} → ∀x y ∈ {x∣ψ})
3	1, 2	cleqf 1167	. 2 ⊢ ({x∣φ} = {x∣ψ} ↔ ∀x(x ∈ {x∣φ} ↔ x ∈ {x∣ψ}))
4		abid 1094	. . . 4 ⊢ (x ∈ {x∣φ} ↔ φ)
5		abid 1094	. . . 4 ⊢ (x ∈ {x∣ψ} ↔ ψ)
6	4, 5	bibi12i 462	. . 3 ⊢ ((x ∈ {x∣φ} ↔ x ∈ {x∣ψ}) ↔ (φ ↔ ψ))
7	6	bial 695	. 2 ⊢ (∀x(x ∈ {x∣φ} ↔ x ∈ {x∣ψ}) ↔ ∀x(φ ↔ ψ))
8	3, 7	bitr 151	1 ⊢ ({x∣φ} = {x∣ψ} ↔ ∀x(φ ↔ ψ))

Syntax hints:   ↔ wb 127  ∀wal 672  {cab 1090   = wceq 1091   ∈ wcel 1092

This theorem is referenced by:  biabi 1181  biabd 1182  pw2en 3348  karden 3551

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099

metamath.org