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Theorem fvmptd4 6959
Description: Deduction version of fvmpt 6935 (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 6942 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2111  cmpt 5174  cfv 6487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5236  ax-nul 5246  ax-pr 5372
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-iota 6443  df-fun 6489  df-fv 6495
This theorem is referenced by:  selvval  22056  mhpmulcl  22070  psdval  22080  psdcoef  22081  evl1deg1  33546  evl1deg2  33547  evl1deg3  33548  mplvrpmrhm  33584  esplyfval  33593  extdgfialglem2  33713  2sqr3minply  33800  cos9thpiminply  33808  evlsvval  42659  evlsvvval  42662  evlsvarval  42664  selvvvval  42684  prjcrvval  42731  dvnprodlem1  46049
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