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Theorem fvmptd3f 7030
Description: Alternate deduction version of fvmpt 7015 with three nonfreeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
Hypotheses
Ref Expression
fvmptd2f.1 (𝜑𝐴𝐷)
fvmptd2f.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptd2f.3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
fvmptd3f.4 𝑥𝐹
fvmptd3f.5 𝑥𝜓
fvmptd3f.6 𝑥𝜑
Assertion
Ref Expression
fvmptd3f (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝜓(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptd3f
StepHypRef Expression
1 fvmptd3f.6 . 2 𝑥𝜑
2 fvmptd3f.4 . . . 4 𝑥𝐹
3 nfmpt1 5255 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2916 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptd3f.5 . . 3 𝑥𝜓
64, 5nfim 1893 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptd2f.1 . . . 4 (𝜑𝐴𝐷)
87elexd 3501 . . 3 (𝜑𝐴 ∈ V)
9 isset 3491 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 218 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 fveq1 6905 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
12 simpr 484 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1312fveq2d 6910 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
147adantr 480 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1512, 14eqeltrd 2838 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
16 fvmptd2f.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
17 eqid 2734 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1817fvmpt2 7026 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1915, 16, 18syl2anc 584 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2013, 19eqtr3d 2776 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2120eqeq2d 2745 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
22 fvmptd2f.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2321, 22sylbid 240 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2411, 23syl5 34 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
251, 6, 10, 24exlimdd 2217 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wex 1775  wnf 1779  wcel 2105  wnfc 2887  Vcvv 3477  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-in 3969  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-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fv 6570
This theorem is referenced by:  fvmptd2f  7031
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