Theorem cbvoprab2 5569

Description: Change the second bound variable in an operation abstraction. (Contributed by Jeff Madsen, 11-Jun-2010.) (Revised by Mario Carneiro, 11-Dec-2016.)

Hypotheses

Ref	Expression
cbvoprab2.1	⊢ Ⅎwφ
cbvoprab2.2	⊢ Ⅎyψ
cbvoprab2.3	⊢ (y = w → (φ ↔ ψ))

Assertion

Ref	Expression
cbvoprab2	⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, w⟩, z⟩ ∣ ψ}

Distinct variable group:   x,w,y,z

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

Proof of Theorem cbvoprab2

Dummy variable v is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . . 7 ⊢ Ⅎw v = ⟨⟨x, y⟩, z⟩
2		cbvoprab2.1	. . . . . . 7 ⊢ Ⅎwφ
3	1, 2	nfan 1824	. . . . . 6 ⊢ Ⅎw(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
4	3	nfex 1843	. . . . 5 ⊢ Ⅎw∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)
5		nfv 1619	. . . . . . 7 ⊢ Ⅎy v = ⟨⟨x, w⟩, z⟩
6		cbvoprab2.2	. . . . . . 7 ⊢ Ⅎyψ
7	5, 6	nfan 1824	. . . . . 6 ⊢ Ⅎy(v = ⟨⟨x, w⟩, z⟩ ∧ ψ)
8	7	nfex 1843	. . . . 5 ⊢ Ⅎy∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ)
9		opeq2 4580	. . . . . . . . 9 ⊢ (y = w → ⟨x, y⟩ = ⟨x, w⟩)
10	9	opeq1d 4585	. . . . . . . 8 ⊢ (y = w → ⟨⟨x, y⟩, z⟩ = ⟨⟨x, w⟩, z⟩)
11	10	eqeq2d 2364	. . . . . . 7 ⊢ (y = w → (v = ⟨⟨x, y⟩, z⟩ ↔ v = ⟨⟨x, w⟩, z⟩))
12		cbvoprab2.3	. . . . . . 7 ⊢ (y = w → (φ ↔ ψ))
13	11, 12	anbi12d 691	. . . . . 6 ⊢ (y = w → ((v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ (v = ⟨⟨x, w⟩, z⟩ ∧ ψ)))
14	13	exbidv 1626	. . . . 5 ⊢ (y = w → (∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ)))
15	4, 8, 14	cbvex 1985	. . . 4 ⊢ (∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃w∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ))
16	15	exbii 1582	. . 3 ⊢ (∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ) ↔ ∃x∃w∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ))
17	16	abbii 2466	. 2 ⊢ {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)} = {v ∣ ∃x∃w∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ)}
18		df-oprab 5529	. 2 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {v ∣ ∃x∃y∃z(v = ⟨⟨x, y⟩, z⟩ ∧ φ)}
19		df-oprab 5529	. 2 ⊢ {⟨⟨x, w⟩, z⟩ ∣ ψ} = {v ∣ ∃x∃w∃z(v = ⟨⟨x, w⟩, z⟩ ∧ ψ)}
20	17, 18, 19	3eqtr4i 2383	1 ⊢ {⟨⟨x, y⟩, z⟩ ∣ φ} = {⟨⟨x, w⟩, z⟩ ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4562  {coprab 5528

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  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-oprab 5529

This theorem is referenced by: (None)