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Theorem fvmptdv 6776
 Description: Alternate deduction version of fvmpt 6759, suitable for iteration. (Contributed by Mario Carneiro, 7-Jan-2017.)
Hypotheses
Ref Expression
fvmptd2f.1 (𝜑𝐴𝐷)
fvmptd2f.2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
fvmptd2f.3 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
Assertion
Ref Expression
fvmptdv (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷   𝜑,𝑥   𝑥,𝐹   𝜓,𝑥
Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem fvmptdv
StepHypRef Expression
1 fvmptd2f.1 . 2 (𝜑𝐴𝐷)
2 fvmptd2f.2 . 2 ((𝜑𝑥 = 𝐴) → 𝐵𝑉)
3 fvmptd2f.3 . 2 ((𝜑𝑥 = 𝐴) → ((𝐹𝐴) = 𝐵𝜓))
4 nfcv 2919 . 2 𝑥𝐹
5 nfv 1915 . 2 𝑥𝜓
61, 2, 3, 4, 5fvmptd2f 6775 1 (𝜑 → (𝐹 = (𝑥𝐷𝐵) → 𝜓))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ↦ cmpt 5112  ‘cfv 6335 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fv 6343 This theorem is referenced by: (None)
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