Theorem abbi 2464

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

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		dfcleq 2347	. 2 ⊢ ({x ∣ φ} = {x ∣ ψ} ↔ ∀y(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}))
2		nfsab1 2343	. . . 4 ⊢ Ⅎx y ∈ {x ∣ φ}
3		nfsab1 2343	. . . 4 ⊢ Ⅎx y ∈ {x ∣ ψ}
4	2, 3	nfbi 1834	. . 3 ⊢ Ⅎx(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ})
5		nfv 1619	. . 3 ⊢ Ⅎy(φ ↔ ψ)
6		df-clab 2340	. . . . 5 ⊢ (y ∈ {x ∣ φ} ↔ [y / x]φ)
7		sbequ12r 1920	. . . . 5 ⊢ (y = x → ([y / x]φ ↔ φ))
8	6, 7	syl5bb 248	. . . 4 ⊢ (y = x → (y ∈ {x ∣ φ} ↔ φ))
9		df-clab 2340	. . . . 5 ⊢ (y ∈ {x ∣ ψ} ↔ [y / x]ψ)
10		sbequ12r 1920	. . . . 5 ⊢ (y = x → ([y / x]ψ ↔ ψ))
11	9, 10	syl5bb 248	. . . 4 ⊢ (y = x → (y ∈ {x ∣ ψ} ↔ ψ))
12	8, 11	bibi12d 312	. . 3 ⊢ (y = x → ((y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}) ↔ (φ ↔ ψ)))
13	4, 5, 12	cbval 1984	. 2 ⊢ (∀y(y ∈ {x ∣ φ} ↔ y ∈ {x ∣ ψ}) ↔ ∀x(φ ↔ ψ))
14	1, 13	bitr2i 241	1 ⊢ (∀x(φ ↔ ψ) ↔ {x ∣ φ} = {x ∣ ψ})

Syntax hints:   ↔ wb 176  ∀wal 1540   = wceq 1642  [wsb 1648   ∈ wcel 1710  {cab 2339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346

This theorem is referenced by:  abbii  2466  abbid  2467  rabbi  2790  dfeu2  4334  dfiota2  4341  iotabi  4349  uniabio  4350  iotanul  4355