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Theorem isfildlem 22393
Description: Lemma for isfild 22394. (Contributed by Mario Carneiro, 1-Dec-2013.)
Hypotheses
Ref Expression
isfild.1 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
isfild.2 (𝜑𝐴 ∈ V)
Assertion
Ref Expression
isfildlem (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥
Allowed substitution hints:   𝜓(𝑥)   𝐵(𝑥)

Proof of Theorem isfildlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3510 . . 3 (𝐵𝐹𝐵 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐵𝐹𝐵 ∈ V))
3 isfild.2 . . . 4 (𝜑𝐴 ∈ V)
4 ssexg 5218 . . . . 5 ((𝐵𝐴𝐴 ∈ V) → 𝐵 ∈ V)
54expcom 414 . . . 4 (𝐴 ∈ V → (𝐵𝐴𝐵 ∈ V))
63, 5syl 17 . . 3 (𝜑 → (𝐵𝐴𝐵 ∈ V))
76adantrd 492 . 2 (𝜑 → ((𝐵𝐴[𝐵 / 𝑥]𝜓) → 𝐵 ∈ V))
8 eleq1 2897 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐹𝐵𝐹))
9 sseq1 3989 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
10 dfsbcq 3771 . . . . . . 7 (𝑦 = 𝐵 → ([𝑦 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
119, 10anbi12d 630 . . . . . 6 (𝑦 = 𝐵 → ((𝑦𝐴[𝑦 / 𝑥]𝜓) ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
128, 11bibi12d 347 . . . . 5 (𝑦 = 𝐵 → ((𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)) ↔ (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
1312imbi2d 342 . . . 4 (𝑦 = 𝐵 → ((𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))) ↔ (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))))
14 nfv 1906 . . . . . 6 𝑥𝜑
15 nfv 1906 . . . . . . 7 𝑥 𝑦𝐹
16 nfv 1906 . . . . . . . 8 𝑥 𝑦𝐴
17 nfsbc1v 3789 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜓
1816, 17nfan 1891 . . . . . . 7 𝑥(𝑦𝐴[𝑦 / 𝑥]𝜓)
1915, 18nfbi 1895 . . . . . 6 𝑥(𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))
2014, 19nfim 1888 . . . . 5 𝑥(𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
21 eleq1 2897 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐹𝑦𝐹))
22 sseq1 3989 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
23 sbceq1a 3780 . . . . . . . 8 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
2422, 23anbi12d 630 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐴𝜓) ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2521, 24bibi12d 347 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐹 ↔ (𝑥𝐴𝜓)) ↔ (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))))
2625imbi2d 342 . . . . 5 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓))) ↔ (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))))
27 isfild.1 . . . . 5 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
2820, 26, 27chvarfv 2232 . . . 4 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2913, 28vtoclg 3565 . . 3 (𝐵 ∈ V → (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
3029com12 32 . 2 (𝜑 → (𝐵 ∈ V → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
312, 7, 30pm5.21ndd 381 1 (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  Vcvv 3492  [wsbc 3769  wss 3933
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-rab 3144  df-v 3494  df-sbc 3770  df-in 3940  df-ss 3949
This theorem is referenced by:  isfild  22394
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