Theorem cbvopab1s 2107

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

Assertion

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

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

Proof of Theorem cbvopab1s

Step	Hyp	Ref	Expression
1		ax-17 925	. . . 4 ⊢ (∃y(w = ⟨x, y⟩ ∧ φ) → ∀z∃y(w = ⟨x, y⟩ ∧ φ))
2		ax-17 925	. . . . . 6 ⊢ (w = ⟨z, y⟩ → ∀x w = ⟨z, y⟩)
3		hbs1 986	. . . . . 6 ⊢ ([z / x]φ → ∀x[z / x]φ)
4	2, 3	hban 704	. . . . 5 ⊢ ((w = ⟨z, y⟩ ∧ [z / x]φ) → ∀x(w = ⟨z, y⟩ ∧ [z / x]φ))
5	4	hbex 701	. . . 4 ⊢ (∃y(w = ⟨z, y⟩ ∧ [z / x]φ) → ∀x∃y(w = ⟨z, y⟩ ∧ [z / x]φ))
6		opeq1 1876	. . . . . . 7 ⊢ (x = z → ⟨x, y⟩ = ⟨z, y⟩)
7	6	cleq2d 1112	. . . . . 6 ⊢ (x = z → (w = ⟨x, y⟩ ↔ w = ⟨z, y⟩))
8		sbequ12 865	. . . . . 6 ⊢ (x = z → (φ ↔ [z / x]φ))
9	7, 8	anbi12d 476	. . . . 5 ⊢ (x = z → ((w = ⟨x, y⟩ ∧ φ) ↔ (w = ⟨z, y⟩ ∧ [z / x]φ)))
10	9	biexdv 936	. . . 4 ⊢ (x = z → (∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃y(w = ⟨z, y⟩ ∧ [z / x]φ)))
11	1, 5, 10	cbvex 849	. . 3 ⊢ (∃x∃y(w = ⟨x, y⟩ ∧ φ) ↔ ∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ))
12	11	biabi 1181	. 2 ⊢ {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ)}
13		df-opab 2098	. 2 Xÿ6; {⟨x, y⟩∣φ} = {w∣∃x∃y(w = ⟨x, y⟩ ∧ φ)}
14		df-opab 2098	. 2 ⊢ {⟨z, y⟩∣[z / x]φ} = {w∣∃z∃y(w = ⟨z, y⟩ ∧ [z / x]φ)}
15	12, 13, 14	3eqtr4 1126	1 ⊢ {⟨x, y⟩∣φ} = {⟨z, y⟩∣[z / x]φ}

Syntax hints:   ∧ wa 196  ∃wex 678   = weq 797  [wsb 852  {cab 1090   = wceq 1091  ⟨cop 1810  {copab 2055

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

metamath.org