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Theorem fvmptd3f 6611
Description: Alternate deduction version of fvmpt 6597 with three non-freeness hypotheses instead of distinct variable conditions. (Contributed by AV, 19-Jan-2022.)
Hypotheses
Ref Expression
fvmptdf.1 (𝜑𝐴𝐷)
fvmptdf.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptdf.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 5026 . . . 4 𝑥(𝑥𝐷𝐵)
42, 3nfeq 2943 . . 3 𝑥 𝐹 = (𝑥𝐷𝐵)
5 fvmptd3f.5 . . 3 𝑥𝜓
64, 5nfim 1859 . 2 𝑥(𝐹 = (𝑥𝐷𝐵) → 𝜓)
7 fvmptdf.1 . . . 4 (𝜑𝐴𝐷)
87elexd 3435 . . 3 (𝜑𝐴 ∈ V)
9 isset 3427 . . 3 (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
108, 9sylib 210 . 2 (𝜑 → ∃𝑥 𝑥 = 𝐴)
11 fveq1 6500 . . 3 (𝐹 = (𝑥𝐷𝐵) → (𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴))
12 simpr 477 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥 = 𝐴)
1312fveq2d 6505 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = ((𝑥𝐷𝐵)‘𝐴))
147adantr 473 . . . . . . . 8 ((𝜑𝑥 = 𝐴) → 𝐴𝐷)
1512, 14eqeltrd 2866 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝑥𝐷)
16 fvmptdf.2 . . . . . . 7 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
17 eqid 2778 . . . . . . . 8 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1817fvmpt2 6607 . . . . . . 7 ((𝑥𝐷𝐵𝑉) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
1915, 16, 18syl2anc 576 . . . . . 6 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝑥) = 𝐵)
2013, 19eqtr3d 2816 . . . . 5 ((𝜑𝑥 = 𝐴) → ((𝑥𝐷𝐵)‘𝐴) = 𝐵)
2120eqeq2d 2788 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) ↔ (𝐹𝐴) = 𝐵))
22 fvmptdf.3 . . . 4 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
2321, 22sylbid 232 . . 3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = ((𝑥𝐷𝐵)‘𝐴) → 𝜓))
2411, 23syl5 34 . 2 ((𝜑𝑥 = 𝐴) → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
251, 6, 10, 24exlimdd 2150 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1507  wex 1742  wnf 1746  wcel 2050  wnfc 2916  Vcvv 3415  cmpt 5009  cfv 6190
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-iota 6154  df-fun 6192  df-fv 6198
This theorem is referenced by:  fvmptdf  6612
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