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Theorem fvmptd3f 7003
Description: Alternate deduction version of fvmpt 6988 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 5246 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2908 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptd3f.5 . . 3 𝑥𝜓
64, 5nfim 1891 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptd2f.1 . . . 4 (𝜑𝐴𝐷)
87elexd 3487 . . 3 (𝜑𝐴 ∈ V)
9 isset 3479 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 217 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 fveq1 6880 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
12 simpr 484 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1312fveq2d 6885 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
147adantr 480 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1512, 14eqeltrd 2825 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
16 fvmptd2f.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
17 eqid 2724 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1817fvmpt2 6999 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1915, 16, 18syl2anc 583 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2013, 19eqtr3d 2766 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2120eqeq2d 2735 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
22 fvmptd2f.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2321, 22sylbid 239 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2411, 23syl5 34 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
251, 6, 10, 24exlimdd 2205 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wex 1773  wnf 1777  wcel 2098  wnfc 2875  Vcvv 3466  cmpt 5221  cfv 6533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fv 6541
This theorem is referenced by:  fvmptd2f  7004
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