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Theorem fvmptd3 39958
Description: Deduction version of fvmpt 6422. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
fvmptd3.1 𝐹 = (𝑥𝐷𝐵)
fvmptd3.2 (𝑥 = 𝐴𝐵 = 𝐶)
fvmptd3.3 (𝜑𝐴𝐷)
fvmptd3.4 (𝜑𝐶𝑉)
Assertion
Ref Expression
fvmptd3 (𝜑 → (𝐹𝐴) = 𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd3
StepHypRef Expression
1 fvmptd3.3 . 2 (𝜑𝐴𝐷)
2 fvmptd3.4 . 2 (𝜑𝐶𝑉)
3 nfcv 2913 . . 3 𝑥𝐴
4 nfcv 2913 . . 3 𝑥𝐶
5 fvmptd3.2 . . 3 (𝑥 = 𝐴𝐵 = 𝐶)
6 fvmptd3.1 . . 3 𝐹 = (𝑥𝐷𝐵)
73, 4, 5, 6fvmptf 6441 . 2 ((𝐴𝐷𝐶𝑉) → (𝐹𝐴) = 𝐶)
81, 2, 7syl2anc 573 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  cmpt 4863  cfv 6029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fv 6037
This theorem is referenced by:  limsuplt2  40496  limsupge  40504  smflimsuplem1  41539  smflimsuplem5  41543  smflimsuplem7  41545
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