Theorem opabbii 4626

Description: Equivalent wff's yield equal class abstractions. (Contributed by NM, 15-May-1995.)

Hypothesis

Ref	Expression
opabbii.1	⊢ (φ ↔ ψ)

Assertion

Ref	Expression
opabbii	⊢ {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ}

Proof of Theorem opabbii

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2353	. 2 ⊢ z = z
2		opabbii.1	. . . 4 ⊢ (φ ↔ ψ)
3	2	a1i 10	. . 3 ⊢ (z = z → (φ ↔ ψ))
4	3	opabbidv 4625	. 2 ⊢ (z = z → {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ})
5	1, 4	ax-mp 5	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ}

Syntax hints:   ↔ wb 176   = wceq 1642  {copab 4622

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  df-opab 4623

This theorem is referenced by:  df1st2  4738  dfsset2  4743  dfco1  4748  dfsi2  4751  fconstopab  4815  xpundi  4832  xpundir  4833  inxp  4863  opabbi2i  4866  cnvco  4894  resopab  4999  opabresid  5003  cnvi  5032  cnvun  5033  cnvxp  5043  coundi  5082  coundir  5083  df2nd2  5111  fvopabg  5391  cbvoprab1  5567  cbvoprab12  5569  dmoprabss  5575  resoprab  5581  fnov  5591  ov6g  5600  mpt2mptx  5708  mptv  5718  scancan  6331