Theorem cbvopab1s 4635

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 31-Jul-2003.)

Assertion

Ref	Expression
cbvopab1s	⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}

Distinct variable groups:   x,y,z   φ,z

Allowed substitution hints:   φ(x,y)

Proof of Theorem cbvopab1s

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . 4 ⊢ Ⅎz∃y(w = ⟨x, y⟩ ∧ φ)
2		nfv 1619	. . . . . 6 ⊢ Ⅎx w = ⟨z, y⟩
3		nfs1v 2106	. . . . . 6 ⊢ Ⅎx[z / x]φ
4	2, 3	nfan 1824	. . . . 5 ⊢ Ⅎx(w = ⟨z, y⟩ ∧ [z / x]φ)
5	4	nfex 1843	. . . 4 ⊢ Ⅎx∃y(w = ⟨z, y⟩ ∧ [z / x]φ)
6		opeq1 4579	. . . . . . 7 ⊢ (x = z → ⟨x, y⟩ = ⟨z, y⟩)
7	6	eqeq2d 2364	. . . . . 6 ⊢ (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
8		sbequ12 1919	. . . . . 6 ⊢ (x = z → (φ ↔ [z / x]φ))
9	7, 8	anbi12d 691	. . . . 5 ⊢ (x = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨z, y⟩ ∧ [z / x]φ)))
10	9	exbidv 1626	. . . 4 ⊢ (x = z → (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃y(w = ⟨z, y⟩ ∧ [z / x]φ)))
11	1, 5, 10	cbvex 1985	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ))
12	11	abbii 2466	. 2 ⊢ {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ)}
13		df-opab 4624	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
14		df-opab 4624	. 2 ⊢ {⟨z, y⟩ ∣ [z / x]φ} = {w ∣ ∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ)}
15	12, 13, 14	3eqtr4i 2383	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ [z / x]φ}

Syntax hints:   ∧ wa 358  ∃wex 1541   = wceq 1642  [wsb 1648  {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: (None)