Theorem rabeqbidv 2732

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 2729	. . 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 2726	. 2 ⊢ (𝜑 → {𝑥 ∈ 𝐵 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})
6	3, 5	eqtrd 2210	1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝜓} = {𝑥 ∈ 𝐵 ∣ 𝜒})

Syntax hints:   → wi 4   ↔ wb 105   = wceq 1353  {crab 2459

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rab 2464

This theorem is referenced by:  elfvmptrab1  5610  mpoxopoveq  6240  supeq123d  6989  phival  12207  dfphi2  12214  ismhm  12807  issubm  12817  issubg  12986  subgex  12989  isnsg  13015  issubrg  13302  cldval  13492  neifval  13533  cnfval  13587  cnpfval  13588  cnprcl2k  13599  hmeofvalg  13696  ispsmet  13716  ismet  13737  isxmet  13738  blfvalps  13778  cncfval  13952