Theorem cbvopab1 2106

Description: Change first bound variable in an ordered pair abstraction, using explicit substitution.

Hypotheses

Ref	Expression
cbvopab1.1	⊢ (φ → ∀zφ)
cbvopab1.2	⊢ (ψ → ∀xψ)
cbvopab1.3	⊢ (x = z → (φ ↔ ψ))

Assertion

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

Distinct variable group(s):   x,y,z

Proof of Theorem cbvopab1

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . 6 ⊢ (w = ⟨x, y⟩ → ∀z w = ⟨x, y⟩)
2		cbvopab1.1	. . . . . 6 ⊢ (φ → ∀zφ)
3	1, 2	hban 704	. . . . 5 ⊢ ((w = ⟨x, y⟩ ∧ φ) → ∀z(w = ⟨x, y⟩ ∧ φ))
4	3	hbex 701	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) → ∀z∃y(w = ⟨x, y⟩ ∧ φ))
5		ax-17 925	. . . . . 6 ⊢ (w = ⟨z, y⟩ → ∀x w = ⟨z, y⟩)
6		cbvopab1.2	. . . . . 6 ⊢ (ψ → ∀xψ)
7	5, 6	hban 704	. . . . 5 ⊢ ((w = ⟨z, y⟩ ∧ ψ) → ∀xÁw = ⟨z, y⟩ ∧ ψ))
8	7	hbex 701	. . . 4 ⊢ (∃y(w = ⟨z, y⟩ ∧ ψ) → ∀x∃y(w = ⟨z, y⟩ ∧ ψ))
9		opeq1 1876	. . . . . . 7 ⊢ (x = z → ⟨x, y⟩ = ⟨z, y⟩)
10	9	cleq2d 1112	. . . . . 6 ⊢ (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
11		cbvopab1.3	. . . . . 6 ⊢ (x = z → (φ ↔ ψ))
12	10, 11	anbi12d 476	. . . . 5 ⊢ (x = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨z, y⟩ ∧ ψ)))
13	12	biexdv 936	. . . 4 ⊢ (x = z → (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃y(w = ⟨z, y⟩ ∧ ψ)))
14	4, 8, 13	cbvex 849	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ ψ))
15	14	biabi 1181	. 2 ⊢ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ ψ)}
16		df-opab 2098	. 2 ⊢ {⟨x, y⟩∣φ} = {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
17		df-opab 2098	. 2 ⊢ {⟨z, y⟩∣ψ} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ ψ)}
18	15, 16, 17	3eqtr4 1126	1 ⊢ {⟨x, y⟩∣φ} = {⟨z, y⟩∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055

This theorem is referenced by:  seqlem1 4662

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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