Theorem cbvopab 4630

Description: Rule used to change bound variables in an ordered-pair class abstraction, using implicit substitution. (Contributed by NM, 14-Sep-2003.)

Hypotheses

Ref	Expression
cbvopab.1	⊢ Ⅎzφ
cbvopab.2	⊢ Ⅎwφ
cbvopab.3	⊢ Ⅎxψ
cbvopab.4	⊢ Ⅎyψ
cbvopab.5	⊢ ((x = z ∧ y = w) → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopab	⊢ {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}

Distinct variable group:   x,y,z,w

Allowed substitution hints:   φ(x,y,z,w)   ψ(x,y,z,w)

Proof of Theorem cbvopab

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . 5 ⊢ Ⅎz v = ⟨x, y⟩
2		cbvopab.1	. . . . 5 ⊢ Ⅎzφ
3	1, 2	nfan 1824	. . . 4 ⊢ Ⅎz(v = ⟨x, y⟩ ∧ φ)
4		nfv 1619	. . . . 5 ⊢ Ⅎw v = ⟨x, y⟩
5		cbvopab.2	. . . . 5 ⊢ Ⅎwφ
6	4, 5	nfan 1824	. . . 4 ⊢ Ⅎw(v = ⟨x, y⟩ ∧ φ)
7		nfv 1619	. . . . 5 ⊢ Ⅎx v = ⟨z, w⟩
8		cbvopab.3	. . . . 5 ⊢ Ⅎxψ
9	7, 8	nfan 1824	. . . 4 ⊢ Ⅎx(v = ⟨z, w⟩ ∧ ψ)
10		nfv 1619	. . . . 5 ⊢ Ⅎy v = ⟨z, w⟩
11		cbvopab.4	. . . . 5 ⊢ Ⅎyψ
12	10, 11	nfan 1824	. . . 4 ⊢ Ⅎy(v = ⟨z, w⟩ ∧ ψ)
13		opeq12 4580	. . . . . 6 ⊢ ((x = z ∧ y = w) → ⟨x, y⟩ = ⟨z, w⟩)
14	13	eqeq2d 2364	. . . . 5 ⊢ ((x = z ∧ y = w) → (v = ⟨x, y⟩ ↔ v = ⟨z, w⟩))
15		cbvopab.5	. . . . 5 ⊢ ((x = z ∧ y = w) → (φ ↔ ψ))
16	14, 15	anbi12d 691	. . . 4 ⊢ ((x = z ∧ y = w) → ((v = ⟨x, y⟩ ∧ φ) ↔ (v = ⟨z, w⟩ ∧ ψ)))
17	3, 6, 9, 12, 16	cbvex2 2005	. . 3 ⊢ (∃x∃y(v = ⟨x, y⟩ ∧ φ) ↔ ∃z∃w(v = ⟨z, w⟩ ∧ ψ))
18	17	abbii 2465	. 2 ⊢ {v ∣ ∃x∃y(v = ⟨x, y⟩ ∧ φ)} = {v ∣ ∃z∃w(v = ⟨z, w⟩ ∧ ψ)}
19		df-opab 4623	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {v ∣ ∃x∃y(v = ⟨x, y⟩ ∧ φ)}
20		df-opab 4623	. 2 ⊢ {⟨z, w⟩ ∣ ψ} = {v ∣ ∃z∃w(v = ⟨z, w⟩ ∧ ψ)}
21	18, 19, 20	3eqtr4i 2383	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨z, w⟩ ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4561  {copab 4622

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-opab 4623

This theorem is referenced by:  cbvopabv  4631