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Theorem raleqbi1dv 2714
Description: Equality deduction for restricted universal quantifier. (Contributed by NM, 16-Nov-1995.)
Hypothesis
Ref Expression
raleqd.1 (𝐴 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
raleqbi1dv (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)

Proof of Theorem raleqbi1dv
StepHypRef Expression
1 raleq 2702 . 2 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜑))
2 raleqd.1 . . 3 (𝐴 = 𝐵 → (𝜑𝜓))
32ralbidv 2506 . 2 (𝐴 = 𝐵 → (∀𝑥𝐵 𝜑 ↔ ∀𝑥𝐵 𝜓))
41, 3bitrd 188 1 (𝐴 = 𝐵 → (∀𝑥𝐴 𝜑 ↔ ∀𝑥𝐵 𝜓))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 105   = wceq 1373  wral 2484
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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489
This theorem is referenced by:  frforeq2  4392  weeq2  4404  peano5  4646  isoeq4  5873  exmidomni  7244  tapeq2  7365  pitonn  7961  peano1nnnn  7965  peano2nnnn  7966  peano5nnnn  8005  peano5nni  9039  1nn  9047  peano2nn  9048  dfuzi  9483  mhmpropd  13298  issubm  13304  isghm  13579  ghmeql  13603  iscmn  13629  dfrhm2  13916  islssm  14119  islssmg  14120  istopg  14471  isbasisg  14516  basis2  14520  eltg2  14525  ispsmet  14795  ismet  14816  isxmet  14817  metrest  14978  cncfval  15044  bj-indeq  15865  bj-nntrans  15887
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