Theorem rabbidv 1809

Description: Equivalent wff's yield equal restricted class abstractions (deduction rule).

Hypothesis

Ref	Expression
rabbidv.1	⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))

Assertion

Ref	Expression
rabbidv	⊢ (φ → {x ∈ A∣ψ} = {x ∈ A∣χ})

Distinct variable group:   φ,x

Proof of Theorem rabbidv

Step	Hyp	Ref	Expression
1		rabbidv.1	. . . 4 ⊢ ((φ ⋀ x ∈ A) → (ψ ↔ χ))
2	1	pm5.32da 651	. . 3 ⊢ (φ → ((x ∈ A ⋀ ψ) ↔ (x ∈ A ⋀ χ)))
3	2	abbidv 1580	. 2 ⊢ (φ → {x∣(x ∈ A ⋀ ψ)} = {x∣(x ∈ A ⋀ χ)})
4		df-rab 1655	. 2 ⊢ {x ∈ A∣ψ} = {x∣(x ∈ A ⋀ ψ)}
5		df-rab 1655	. 2 ⊢ {x ∈ A∣χ} = {x∣(x ∈ A ⋀ χ)}
6	3, 4, 5	3eqtr4g 1534	1 ⊢ (φ → {x ∈ A∣ψ} = {x ∈ A∣χ})

Syntax hints:   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  {cab 1466  {crab 1651

This theorem is referenced by:  rabbisdv 1810  onsucmin 3078  dfinfmr 6069  infmsup 6070  supxrre 6085  iooint 6373  cncnplem4 7774  blin 7849  addinv 8124  ee7.2a 10420

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-rab 1655

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