Theorem cbvoprab3v 3030

Description: Rule used to change the third bound variable in an operation abstraction, using implicit substitution.

Hypothesis

Ref	Expression
cbvoprab3v.1	⊢ (z = w → (φ ↔ ψ))

Assertion

Ref	Expression
cbvoprab3v	⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨⟨x, y⟩, w⟩∣ψ}

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

Proof of Theorem cbvoprab3v

Step	Hyp	Ref	Expression
1		cbvoprab3v.1	. . . . 5 ⊢ (z = w → (φ ↔ ψ))
2	1	anbi2d 468	. . . 4 ⊢ (z = w → ((v = ⟨x, y⟩ ∧ φ) ↔ (v = ⟨x, y⟩ ∧ ψ)))
3	2	bi2exdv 938	. . 3 ⊢ (z = w → (∃x∃y(v = ⟨x, y⟩ ∧ φ) ↔ ∃x∃y(v = ⟨x, y⟩ ∧ ψ)))
4	3	cbvopab2v 2109	. 2 ⊢ {⟨v, z⟩∣∃x∃y(v = ⟨x, y⟩ ∧ φ)} = {⟨v, w⟩∣∃x∃y(v = ⟨x, y⟩ ∧ ψ)}
5		dfoprab2 3021	. 2 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨v, z⟩∣∃x∃y(v = ⟨x, y⟩ ∧ φ)}
6		dfoprab2 3021	. 2 ⊢ {⟨⟨x, y⟩, w⟩∣ψ} = {⟨v, w⟩∣∃x∃y(v = ⟨x, y⟩ ∧ ψ)}
7	4, 5, 6	3eqtr4 1126	1 ⊢ {⟨⟨x, y⟩, z⟩∣φ} = {⟨⟨x, y⟩, w⟩∣ψ}

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∃wex 678   = weq 797   = wceq 1091  ⟨cop 1810  {copab 2055  {copab2 3002

This theorem is referenced by:  ruclem12 4896

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-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

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-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-opab 2098  df-oprab 3004

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