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Theorem fvmptd4 7039
Description: Deduction version of fvmpt 7015 (where the substitution hypothesis does not have the antecedent 𝜑). (Contributed by SN, 26-Jul-2024.)
Hypotheses
Ref Expression
fvmptd4.1 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptd4.2 (𝜑𝐹 = (𝑥𝐷𝐵))
fvmptd4.3 (𝜑𝐴𝐷)
fvmptd4.4 (𝜑𝐶𝑉)
Assertion
Ref Expression
fvmptd4 (𝜑 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝜑,𝑥   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd4
StepHypRef Expression
1 fvmptd4.2 . 2 (𝜑𝐹 = (𝑥𝐷𝐵))
2 fvmptd4.1 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
32adantl 481 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
4 fvmptd4.3 . 2 (𝜑𝐴𝐷)
5 fvmptd4.4 . 2 (𝜑𝐶𝑉)
61, 3, 4, 5fvmptd 7022 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1536  wcel 2105  cmpt 5230  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570
This theorem is referenced by:  selvval  22156  mhpmulcl  22170  psdval  22180  psdcoef  22181  evl1deg1  33580  evl1deg2  33581  evl1deg3  33582  2sqr3minply  33752  evlsvval  42546  evlsvvval  42549  evlsvarval  42551  selvvvval  42571  prjcrvval  42618  dvnprodlem1  45901  uspgrimprop  47810
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