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Theorem eqrelrdv 4852
 Description: Deduce equality of relations from equivalence of membership. (Contributed by Rodolfo Medina, 10-Oct-2010.) (Revised by Scott Fenton, 16-Apr-2021.)
Hypothesis
Ref Expression
eqrelrdv.1 (φ → (x, y Ax, y B))
Assertion
Ref Expression
eqrelrdv (φA = B)
Distinct variable groups:   x,y,A   x,B,y   φ,x,y

Proof of Theorem eqrelrdv
StepHypRef Expression
1 eqrelrdv.1 . . 3 (φ → (x, y Ax, y B))
21alrimivv 1632 . 2 (φxy(x, y Ax, y B))
3 eqrel 4845 . 2 (A = Bxy(x, y Ax, y B))
42, 3sylibr 203 1 (φA = B)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   = wceq 1642   ∈ wcel 1710  ⟨cop 4561 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-0c 4377  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568 This theorem is referenced by:  iss  5000  funssres  5144  fcnvres  5243  dffn5  5363  fnasrn  5417  fsn  5432
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