Theorem opabbii 3815

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 2037	. 2 ⊢ z = z
2		opabbii.1	. . . 4 ⊢ (φ ↔ ψ)
3	2	a1i 9	. . 3 ⊢ (z = z → (φ ↔ ψ))
4	3	opabbidv 3814	. 2 ⊢ (z = z → {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ})
5	1, 4	ax-mp 7	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨x, y⟩ ∣ ψ}

Syntax hints:   ↔ wb 98   = wceq 1242  {copab 3808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-opab 3810

This theorem is referenced by:  mptv  3844  fconstmpt  4330  xpundi  4339  xpundir  4340  inxp  4413  cnvco  4463  resopab  4595  opabresid  4602  cnvi  4671  cnvun  4672  cnvin  4674  cnvxp  4685  cnvcnv3  4713  coundi  4765  coundir  4766  mptun  4972  fvopab6  5207  cbvoprab1  5518  cbvoprab12  5520  dmoprabss  5528  mpt2mptx  5537  resoprab  5539  ov6g  5580  dfoprab3s  5758  dfoprab3  5759  dfoprab4  5760  xpcomco  6236  dmaddpq  6363  dmmulpq  6364  recmulnqg  6375  enq0enq  6414  ltrelxr  6877  ltxr  8465