MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isfildlem Structured version   Visualization version   GIF version

Theorem isfildlem 21571
Description: Lemma for isfild 21572. (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 3198 . . 3 (𝐵𝐹𝐵 ∈ V)
21a1i 11 . 2 (𝜑 → (𝐵𝐹𝐵 ∈ V))
3 isfild.2 . . . 4 (𝜑𝐴 ∈ V)
4 ssexg 4764 . . . . 5 ((𝐵𝐴𝐴 ∈ V) → 𝐵 ∈ V)
54expcom 451 . . . 4 (𝐴 ∈ V → (𝐵𝐴𝐵 ∈ V))
63, 5syl 17 . . 3 (𝜑 → (𝐵𝐴𝐵 ∈ V))
76adantrd 484 . 2 (𝜑 → ((𝐵𝐴[𝐵 / 𝑥]𝜓) → 𝐵 ∈ V))
8 eleq1 2686 . . . . . 6 (𝑦 = 𝐵 → (𝑦𝐹𝐵𝐹))
9 sseq1 3605 . . . . . . 7 (𝑦 = 𝐵 → (𝑦𝐴𝐵𝐴))
10 dfsbcq 3419 . . . . . . 7 (𝑦 = 𝐵 → ([𝑦 / 𝑥]𝜓[𝐵 / 𝑥]𝜓))
119, 10anbi12d 746 . . . . . 6 (𝑦 = 𝐵 → ((𝑦𝐴[𝑦 / 𝑥]𝜓) ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
128, 11bibi12d 335 . . . . 5 (𝑦 = 𝐵 → ((𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)) ↔ (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
1312imbi2d 330 . . . 4 (𝑦 = 𝐵 → ((𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))) ↔ (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))))
14 nfv 1840 . . . . . 6 𝑥𝜑
15 nfv 1840 . . . . . . 7 𝑥 𝑦𝐹
16 nfv 1840 . . . . . . . 8 𝑥 𝑦𝐴
17 nfsbc1v 3437 . . . . . . . 8 𝑥[𝑦 / 𝑥]𝜓
1816, 17nfan 1825 . . . . . . 7 𝑥(𝑦𝐴[𝑦 / 𝑥]𝜓)
1915, 18nfbi 1830 . . . . . 6 𝑥(𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))
2014, 19nfim 1822 . . . . 5 𝑥(𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
21 eleq1 2686 . . . . . . 7 (𝑥 = 𝑦 → (𝑥𝐹𝑦𝐹))
22 sseq1 3605 . . . . . . . 8 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
23 sbceq1a 3428 . . . . . . . 8 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
2422, 23anbi12d 746 . . . . . . 7 (𝑥 = 𝑦 → ((𝑥𝐴𝜓) ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2521, 24bibi12d 335 . . . . . 6 (𝑥 = 𝑦 → ((𝑥𝐹 ↔ (𝑥𝐴𝜓)) ↔ (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓))))
2625imbi2d 330 . . . . 5 (𝑥 = 𝑦 → ((𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓))) ↔ (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))))
27 isfild.1 . . . . 5 (𝜑 → (𝑥𝐹 ↔ (𝑥𝐴𝜓)))
2820, 26, 27chvar 2261 . . . 4 (𝜑 → (𝑦𝐹 ↔ (𝑦𝐴[𝑦 / 𝑥]𝜓)))
2913, 28vtoclg 3252 . . 3 (𝐵 ∈ V → (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
3029com12 32 . 2 (𝜑 → (𝐵 ∈ V → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓))))
312, 7, 30pm5.21ndd 369 1 (𝜑 → (𝐵𝐹 ↔ (𝐵𝐴[𝐵 / 𝑥]𝜓)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  [wsbc 3417  wss 3555
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-sbc 3418  df-in 3562  df-ss 3569
This theorem is referenced by:  isfild  21572
  Copyright terms: Public domain W3C validator