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Theorem dfon2lem1 31989
Description: Lemma for dfon2 31998. (Contributed by Scott Fenton, 28-Feb-2011.)
Assertion
Ref Expression
dfon2lem1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}

Proof of Theorem dfon2lem1
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 truni 4915 . 2 (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}Tr 𝑦 → Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)})
2 nfsbc1v 3592 . . . . 5 𝑥[𝑦 / 𝑥]𝜑
3 nfv 1988 . . . . 5 𝑥Tr 𝑦
4 nfsbc1v 3592 . . . . 5 𝑥[𝑦 / 𝑥]𝜓
52, 3, 4nf3an 1976 . . . 4 𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)
6 vex 3339 . . . 4 𝑦 ∈ V
7 sbceq1a 3583 . . . . 5 (𝑥 = 𝑦 → (𝜑[𝑦 / 𝑥]𝜑))
8 treq 4906 . . . . 5 (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9 sbceq1a 3583 . . . . 5 (𝑥 = 𝑦 → (𝜓[𝑦 / 𝑥]𝜓))
107, 8, 93anbi123d 1544 . . . 4 (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓)))
115, 6, 10elabf 3485 . . 3 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦[𝑦 / 𝑥]𝜓))
1211simp2bi 1141 . 2 (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)} → Tr 𝑦)
131, 12mprg 3060 1 Tr {𝑥 ∣ (𝜑 ∧ Tr 𝑥𝜓)}
Colors of variables: wff setvar class
Syntax hints:  w3a 1072  wcel 2135  {cab 2742  [wsbc 3572   cuni 4584  Tr wtr 4900
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ral 3051  df-rex 3052  df-v 3338  df-sbc 3573  df-in 3718  df-ss 3725  df-uni 4585  df-iun 4670  df-tr 4901
This theorem is referenced by:  dfon2lem3  31991  dfon2lem7  31995
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