Theorem cbvopab1 4632

Description: Change first bound variable in an ordered-pair class abstraction, using explicit substitution. (Contributed by NM, 6-Oct-2004.) (Revised by Mario Carneiro, 14-Oct-2016.)

Hypotheses

Ref	Expression
cbvopab1.1	⊢ Ⅎzφ
cbvopab1.2	⊢ Ⅎxψ
cbvopab1.3	⊢ (x = z → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopab1	⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Distinct variable groups:   x,y   y,z

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

Proof of Theorem cbvopab1

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1619	. . . . 5 ⊢ Ⅎv∃y(w = ⟨x, y⟩ ∧ φ)
2		nfv 1619	. . . . . . 7 ⊢ Ⅎx w = ⟨v, y⟩
3		nfs1v 2106	. . . . . . 7 ⊢ Ⅎx[v / x]φ
4	2, 3	nfan 1824	. . . . . 6 ⊢ Ⅎx(w = ⟨v, y⟩ ∧ [v / x]φ)
5	4	nfex 1843	. . . . 5 ⊢ Ⅎx∃y(w = ⟨v, y⟩ ∧ [v / x]φ)
6		opeq1 4578	. . . . . . . 8 ⊢ (x = v → ⟨x, y⟩ = ⟨v, y⟩)
7	6	eqeq2d 2364	. . . . . . 7 ⊢ (x = v → (w = ⟨x, y⟩ ↔ w = ⟨v, y⟩))
8		sbequ12 1919	. . . . . . 7 ⊢ (x = v → (φ ↔ [v / x]φ))
9	7, 8	anbi12d 691	. . . . . 6 ⊢ (x = v → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨v, y⟩ ∧ [v / x]φ)))
10	9	exbidv 1626	. . . . 5 ⊢ (x = v → (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃y(w = ⟨v, y⟩ ∧ [v / x]φ)))
11	1, 5, 10	cbvex 1985	. . . 4 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃v∃y(w = ⟨v, y⟩ ∧ [v / x]φ))
12		nfv 1619	. . . . . . 7 ⊢ Ⅎz w = ⟨v, y⟩
13		cbvopab1.1	. . . . . . . 8 ⊢ Ⅎzφ
14	13	nfsb 2109	. . . . . . 7 ⊢ Ⅎz[v / x]φ
15	12, 14	nfan 1824	. . . . . 6 ⊢ Ⅎz(w = ⟨v, y⟩ ∧ [v / x]φ)
16	15	nfex 1843	. . . . 5 ⊢ Ⅎz∃y(w = ⟨v, y⟩ ∧ [v / x]φ)
17		nfv 1619	. . . . 5 ⊢ Ⅎv∃y(w = ⟨z, y⟩ ∧ ψ)
18		opeq1 4578	. . . . . . . 8 ⊢ (v = z → ⟨v, y⟩ = ⟨z, y⟩)
19	18	eqeq2d 2364	. . . . . . 7 ⊢ (v = z → (w = ⟨v, y⟩ ↔ w = ⟨z, y⟩))
20		sbequ 2060	. . . . . . . 8 ⊢ (v = z → ([v / x]φ ↔ [z / x]φ))
21		cbvopab1.2	. . . . . . . . 9 ⊢ Ⅎxψ
22		cbvopab1.3	. . . . . . . . 9 ⊢ (x = z → (φ ↔ ψ))
23	21, 22	sbie 2038	. . . . . . . 8 ⊢ ([z / x]φ ↔ ψ)
24	20, 23	syl6bb 252	. . . . . . 7 ⊢ (v = z → ([v / x]φ ↔ ψ))
25	19, 24	anbi12d 691	. . . . . 6 ⊢ (v = z → ((w = ⟨v, y⟩ ∧ [v / x]φ) ↔ (w = ⟨z, y⟩ ∧ ψ)))
26	25	exbidv 1626	. . . . 5 ⊢ (v = z → (∃y(w = ⟨v, y⟩ ∧ [v / x]φ) ↔ ∃y(w = ⟨z, y⟩ ∧ ψ)))
27	16, 17, 26	cbvex 1985	. . . 4 ⊢ (∃v∃y(w = ⟨v, y⟩ ∧ [v / x]φ) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ ψ))
28	11, 27	bitri 240	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ ψ))
29	28	abbii 2465	. 2 ⊢ {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃z∃y(w = ⟨z, y⟩ ∧ ψ)}
30		df-opab 4623	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
31		df-opab 4623	. 2 ⊢ {⟨z, y⟩ ∣ ψ} = {w ∣ ∃z∃y(w = ⟨z, y⟩ ∧ ψ)}
32	29, 30, 31	3eqtr4i 2383	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨z, y⟩ ∣ ψ}

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541  Ⅎwnf 1544   = wceq 1642  [wsb 1648  {cab 2339  ⟨cop 4561  {copab 4622

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 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:  cbvopab1v  4635  cbvmpt  5676