Theorem cbvopab2v 2109

Description: Rule used to change the second bound variable in an ordered pair abstraction, using implicit substitution.

Hypothesis

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

Assertion

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

Distinct variable group(s):   x,y,z   φ,z   ψ,y

Proof of Theorem cbvopab2v

Step	Hyp	Ref	Expression
1		opeq2 1877	. . . . . . 7 ⊢ (y = z → ⟨x, y⟩ = ⟨x, z⟩)
2	1	cleq2d 1112	. . . . . 6 ⊢ (y = z → (w = ⟨x, y⟩ ↔ w = ⟨x, z⟩))
3		cbvopab2v.1	. . . . . 6 ⊢ (y = z → (φ ↔ ψ))
4	2, 3	anbi12d 476	. . . . 5 ⊢ (y = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨x, z⟩ ∧ ψ)))
5	4	cbvexv 973	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z(w = ⟨x, z⟩ ∧ ψ))
6	5	biex 733	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃x∃z(w = ⟨x, z⟩ ∧ ψ))
7	6	biabi 1181	. 2 ⊢ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w∣∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
8		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣φ} = {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
9		df-opab 2098	. 2 ⊢ {⟨x, z⟩∣ψ} = {w∣∃x∃z(w = ⟨x, z⟩ ∧ ψ)}
10	7, 8, 9	3eqtr4 1126	1 ⊢ {⟨x, y⟩∣φ} = {⟨x, z⟩∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055

This theorem is referenced by:  cbvoprab3v 3030  ac6 3576

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098

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