Theorem rabeqbidv 2721

Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)

Hypotheses

Ref	Expression
rabeqbidv.1	⊢ (𝜑 → 𝐴 = 𝐵)
rabeqbidv.2	⊢ (𝜑 → (𝜓 ↔ 𝜒))

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝜑,𝑥

Allowed substitution hints:   𝜓(𝑥)   𝜒(𝑥)

Proof of Theorem rabeqbidv

Step	Hyp	Ref	Expression
1		rabeqbidv.1	. . 3 ⊢ (𝜑 → 𝐴 = 𝐵)
2		rabeq 2718	. . 3 ⊢ (𝐴 = 𝐵 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
3	1, 2	syl 14	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜓})
4		rabeqbidv.2	. . 3 ⊢ (𝜑 → (𝜓 ↔ 𝜒))
5	4	rabbidv 2715	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
6	3, 5	eqtrd 2198	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 104   = wceq 1343  {crab 2448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rab 2453

This theorem is referenced by:  elfvmptrab1  5580  mpoxopoveq  6208  supeq123d  6956  phival  12145  dfphi2  12152  cldval  12739  neifval  12780  cnfval  12834  cnpfval  12835  cnprcl2k  12846  hmeofvalg  12943  ispsmet  12963  ismet  12984  isxmet  12985  blfvalps  13025  cncfval  13199