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Theorem rabeqbidv 2546
Description: Equality of restricted class abstractions. (Contributed by Jeff Madsen, 1-Dec-2009.)
Hypotheses
Ref Expression
rabeqbidv.1 (φA = B)
rabeqbidv.2 (φ → (ψχ))
Assertion
Ref Expression
rabeqbidv (φ → {x Aψ} = {x Bχ})
Distinct variable groups:   x,A   x,B   φ,x
Allowed substitution hints:   ψ(x)   χ(x)

Proof of Theorem rabeqbidv
StepHypRef Expression
1 rabeqbidv.1 . . 3 (φA = B)
2 rabeq 2545 . . 3 (A = B → {x Aψ} = {x Bψ})
31, 2syl 14 . 2 (φ → {x Aψ} = {x Bψ})
4 rabeqbidv.2 . . 3 (φ → (ψχ))
54rabbidv 2543 . 2 (φ → {x Bψ} = {x Bχ})
63, 5eqtrd 2069 1 (φ → {x Aψ} = {x Bχ})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 98   = wceq 1242  {crab 2304
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rab 2309
This theorem is referenced by:  mpt2xopoveq  5796
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