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Theorem fvmptd4 39792
Description: Deduction version of fvmpt 6775 (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 485 . 2 ((𝜑𝑥 = 𝐴) → 𝐵 = 𝐶)
4 fvmptd4.3 . 2 (𝜑𝐴𝐷)
5 fvmptd4.4 . 2 (𝜑𝐶𝑉)
61, 3, 4, 5fvmptd 6782 1 (𝜑 → (𝐹𝐴) = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  cmpt 5110  cfv 6339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347
This theorem is referenced by:  evlsvarval  39853  evlsbagval  39854
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