Theorem cbvopab2v 3825

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution. (Contributed by NM, 2-Sep-1999.)

Hypothesis

Ref	Expression
cbvopab2v.1	⊢ (y = z → (φ ↔ ψ))

Assertion

Ref	Expression
cbvopab2v	⊢ {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

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

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

Proof of Theorem cbvopab2v

Dummy variable w is distinct from all other variables.

Step	Hyp	Ref	Expression
1		opeq2 3541	. . . . . . 7 ⊢ (y = z → ⟨x, y⟩ = ⟨x, z⟩)
2	1	eqeq2d 2048	. . . . . 6 ⊢ (y = z → (w = ⟨x, y⟩ ↔ w = ⟨x, z⟩))
3		cbvopab2v.1	. . . . . 6 ⊢ (y = z → (φ ↔ ψ))
4	2, 3	anbi12d 442	. . . . 5 ⊢ (y = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨x, z⟩ ∧ ψ)))
5	4	cbvexv 1792	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z(w = ⟨x, z⟩ ∧ ψ))
6	5	exbii 1493	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃z(w = ⟨x, z⟩ ∧ ψ))
7	6	abbii 2150	. 2 ⊢ {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
8		df-opab 3810	. 2 ⊢ {⟨x, y⟩ ∣ φ} = {w ∣ ∃x∃y(w = ⟨x, y⟩ ∧ φ)}
9		df-opab 3810	. 2 ⊢ {⟨x, z⟩ ∣ ψ} = {w ∣ ∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
10	7, 8, 9	3eqtr4i 2067	1 ⊢ {⟨x, y⟩ ∣ φ} = {⟨x, z⟩ ∣ ψ}

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378  {cab 2023  ⟨cop 3370  {copab 3808

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810

This theorem is referenced by: (None)