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Theorem spcgft 3503
Description: A closed version of spcgf 3506. (Contributed by Andrew Salmon, 6-Jun-2011.) (Revised by Mario Carneiro, 4-Jan-2017.)
Hypotheses
Ref Expression
spcimgft.1 𝑥𝜓
spcimgft.2 𝑥𝐴
Assertion
Ref Expression
spcgft (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))

Proof of Theorem spcgft
StepHypRef Expression
1 biimp 218 . . . 4 ((𝜑𝜓) → (𝜑𝜓))
21imim2i 16 . . 3 ((𝑥 = 𝐴 → (𝜑𝜓)) → (𝑥 = 𝐴 → (𝜑𝜓)))
32alimi 1819 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → ∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)))
4 spcimgft.1 . . 3 𝑥𝜓
5 spcimgft.2 . . 3 𝑥𝐴
64, 5spcimgft 3502 . 2 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
73, 6syl 17 1 (∀𝑥(𝑥 = 𝐴 → (𝜑𝜓)) → (𝐴𝐵 → (∀𝑥𝜑𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wal 1541   = wceq 1543  wnf 1791  wcel 2110  wnfc 2884
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-v 3410
This theorem is referenced by:  spcgf  3506  rspct  3523
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