Theorem cbvopab2 4634

Description: Change second bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 22-Aug-2013.)

Hypotheses

Ref	Expression
cbvopab2.1	⊢ Ⅎzφ
cbvopab2.2	⊢ Ⅎyψ
cbvopab2.3	⊢ (y = z → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopab2	⊢ {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

Distinct variable group:   x,y,z

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

Proof of Theorem cbvopab2

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . . 6 ⊢ Ⅎz w = ⟨x, y⟩
2		cbvopab2.1	. . . . . 6 ⊢ Ⅎzφ
3	1, 2	nfan 1824	. . . . 5 ⊢ Ⅎz(w = ⟨x, y⟩ ∧ φ)
4		nfv 1619	. . . . . 6 ⊢ Ⅎy w = ⟨x, z⟩
5		cbvopab2.2	. . . . . 6 ⊢ Ⅎyψ
6	4, 5	nfan 1824	. . . . 5 ⊢ Ⅎy(w = ⟨x, z⟩ ∧ ψ)
7		opeq2 4580	. . . . . . 7 ⊢ (y = z → ⟨x, y⟩ = ⟨x, z⟩)
8	7	eqeq2d 2364	. . . . . 6 ⊢ (y = z → (w = ⟨x, y⟩ ↔ w = ⟨x, z⟩))
9		cbvopab2.3	. . . . . 6 ⊢ (y = z → (φ ↔ ψ))
10	8, 9	anbi12d 691	. . . . 5 ⊢ (y = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨x, z⟩ ∧ ψ)))
11	3, 6, 10	cbvex 1985	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z(w = ⟨x, z⟩ ∧ ψ))
12	11	exbii 1582	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃z(w = ⟨x, z⟩ ∧ ψ))
13	12	abbii 2466	. 2 ⊢ {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
14		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
15		df-opab 4624	. 2 ⊢ {⟨x, z⟩ ∣ ψ} = {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
16	13, 14, 15	3eqtr4i 2383	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  {cab 2339  ⟨cop 4562  {copab 4623

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-opab 4624

This theorem is referenced by:  cbvoprab3  5572