Theorem opabbii 4049

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 2165	. 2 ⊢ 𝑧 = 𝑧
2		opabbii.1	. . . 4 ⊢ (𝜑 ↔ 𝜓)
3	2	a1i 9	. . 3 ⊢ (𝑧 = 𝑧 → (𝜑 ↔ 𝜓))
4	3	opabbidv 4048	. 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 104   = wceq 1343  {copab 4042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-opab 4044

This theorem is referenced by:  mptv  4079  fconstmpt  4651  xpundi  4660  xpundir  4661  inxp  4738  cnvco  4789  resopab  4928  opabresid  4937  cnvi  5008  cnvun  5009  cnvin  5011  cnvxp  5022  cnvcnv3  5053  coundi  5105  coundir  5106  mptun  5319  fvopab6  5582  cbvoprab1  5914  cbvoprab12  5916  dmoprabss  5924  mpomptx  5933  resoprab  5938  ov6g  5979  dfoprab3s  6158  dfoprab3  6159  dfoprab4  6160  mapsncnv  6661  xpcomco  6792  dmaddpq  7320  dmmulpq  7321  recmulnqg  7332  enq0enq  7372  ltrelxr  7959  ltxr  9711  shftidt2  10774  lmfval  12842  lmbr  12863  cnmptid  12931