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Theorem fvmptd4 7032
Description: Deduction version of fvmpt 7008 (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 480 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
4 fvmptd4.3 . 2 (𝜑𝐴𝐷)
5 fvmptd4.4 . 2 (𝜑𝐶𝑉)
61, 3, 4, 5fvmptd 7015 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  cmpt 5233  cfv 6551
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 2166  ax-ext 2698  ax-sep 5301  ax-nul 5308  ax-pr 5431
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-mpt 5234  df-id 5578  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-iota 6503  df-fun 6553  df-fv 6559
This theorem is referenced by:  evlsvval  41796  evlsvvval  41799  evlsvarval  41801  selvvvval  41821  prjcrvval  42059  uspgrimprop  47222
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