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Theorem isfildlem 22438
Description: Lemma for isfild 22439. (Contributed by Mario Carneiro, 1-Dec-2013.)
Hypotheses
Ref Expression
isfild.1 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
isfild.2 (𝜑𝐴𝑉)
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)   𝑉(𝑥)

Proof of Theorem isfildlem
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3491 . . 3 (𝐵𝐹𝐵 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐵𝐹𝐵 ∈ V))
3 isfild.2 . . . 4 (𝜑𝐴𝑉)
4 ssexg 5201 . . . . 5 ((𝐵𝐴𝐴𝑉) → 𝐵 ∈ V)
54expcom 416 . . . 4 (𝐴𝑉 → (𝐵𝐴𝐵 ∈ V))
63, 5syl 17 . . 3 (𝜑 → (𝐵𝐴𝐵 ∈ V))
76adantrd 494 . 2 (𝜑 → ((𝐵𝐴[𝐵 / 𝑥]𝜓) → 𝐵 ∈ V))
8 eleq1 2898 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐹𝐵𝐹))
9 sseq1 3968 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
10 dfsbcq 3753 . . . . . . 7 (𝑦 = 𝐵 → ([𝑦 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
119, 10anbi12d 632 . . . . . 6 (𝑦 = 𝐵 → ((𝑦𝐴[𝑦 / 𝑥]𝜓) ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
128, 11bibi12d 348 . . . . 5 (𝑦 = 𝐵 → ((𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)) ↔ (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
1312imbi2d 343 . . . 4 (𝑦 = 𝐵 → ((𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))) ↔ (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))))
14 nfv 1915 . . . . . 6 𝑥𝜑
15 nfv 1915 . . . . . . 7 𝑥 𝑦𝐹
16 nfv 1915 . . . . . . . 8 𝑥 𝑦𝐴
17 nfsbc1v 3771 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜓
1816, 17nfan 1900 . . . . . . 7 𝑥(𝑦𝐴[𝑦 / 𝑥]𝜓)
1915, 18nfbi 1904 . . . . . 6 𝑥(𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))
2014, 19nfim 1897 . . . . 5 𝑥(𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
21 eleq1 2898 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐹𝑦𝐹))
22 sseq1 3968 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
23 sbceq1a 3762 . . . . . . . 8 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
2422, 23anbi12d 632 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐴𝜓) ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2521, 24bibi12d 348 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐹 ↔ (𝑥𝐴𝜓)) ↔ (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))))
2625imbi2d 343 . . . . 5 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓))) ↔ (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))))
27 isfild.1 . . . . 5 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
2820, 26, 27chvarfv 2242 . . . 4 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2913, 28vtoclg 3546 . . 3 (𝐵 ∈ V → (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
3029com12 32 . 2 (𝜑 → (𝐵 ∈ V → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
312, 7, 30pm5.21ndd 383 1 (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208  wa 398   = wceq 1537  wcel 2114  Vcvv 3473  [wsbc 3751  wss 3912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5177
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-rab 3134  df-v 3475  df-sbc 3752  df-in 3919  df-ss 3928
This theorem is referenced by:  isfild  22439
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