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Theorem funALTVeq 37212
Description: Equality theorem for function predicate. (Contributed by NM, 16-Aug-1994.)
Assertion
Ref Expression
funALTVeq (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))

Proof of Theorem funALTVeq
StepHypRef Expression
1 eqimss2 4005 . . 3 (𝐴 = 𝐵𝐵𝐴)
2 funALTVss 37211 . . 3 (𝐵𝐴 → ( FunALTV 𝐴 → FunALTV 𝐵))
31, 2syl 17 . 2 (𝐴 = 𝐵 → ( FunALTV 𝐴 → FunALTV 𝐵))
4 eqimss 4004 . . 3 (𝐴 = 𝐵𝐴𝐵)
5 funALTVss 37211 . . 3 (𝐴𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
64, 5syl 17 . 2 (𝐴 = 𝐵 → ( FunALTV 𝐵 → FunALTV 𝐴))
73, 6impbid 211 1 (𝐴 = 𝐵 → ( FunALTV 𝐴 ↔ FunALTV 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wss 3914   FunALTV wfunALTV 36715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-br 5110  df-opab 5172  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-coss 36923  df-cnvrefrel 37039  df-funALTV 37194
This theorem is referenced by:  funALTVeqi  37213  funALTVeqd  37214
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