Theorem opabbii 4096

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 2193	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4095	. 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 105   = wceq 1364  {copab 4089

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-opab 4091

This theorem is referenced by:  mptv  4126  fconstmpt  4706  xpundi  4715  xpundir  4716  inxp  4796  cnvco  4847  resopab  4986  opabresid  4995  cnvi  5070  cnvun  5071  cnvin  5073  cnvxp  5084  cnvcnv3  5115  coundi  5167  coundir  5168  mptun  5385  fvopab6  5654  cbvoprab1  5990  cbvoprab12  5992  dmoprabss  6000  mpomptx  6009  resoprab  6014  ov6g  6056  dfoprab3s  6243  dfoprab3  6244  dfoprab4  6245  mapsncnv  6749  xpcomco  6880  dmaddpq  7439  dmmulpq  7440  recmulnqg  7451  enq0enq  7491  ltrelxr  8080  ltxr  9841  shftidt2  10976  prdsex  12880  releqgg  13290  eqgex  13291  dvdsrzring  14091  lmfval  14360  lmbr  14381  cnmptid  14449