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Metamath Proof Explorer

Theorem dfon2lem1

Description: Lemma for dfon2 . (Contributed by Scott Fenton, 28-Feb-2011)

Ref Expression
Assertion dfon2lem1 Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) }

Proof

Step Hyp Ref Expression
1 truni ⊢ ( ∀ 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } Tr 𝑦 → Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } )
2 nfsbc1v ⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜑
3 nfv ⊢ Ⅎ 𝑥 Tr 𝑦
4 nfsbc1v ⊢ Ⅎ 𝑥 [ 𝑦 / 𝑥 ] 𝜓
5 2 3 4 nf3an ⊢ Ⅎ 𝑥 ( [ 𝑦 / 𝑥 ] 𝜑 ∧ Tr 𝑦 ∧ [ 𝑦 / 𝑥 ] 𝜓 )
6 vex ⊢ 𝑦 ∈ V
7 sbceq1a ⊢ ( 𝑥 = 𝑦 → ( 𝜑 ↔ [ 𝑦 / 𝑥 ] 𝜑 ) )
8 treq ⊢ ( 𝑥 = 𝑦 → ( Tr 𝑥 ↔ Tr 𝑦 ) )
9 sbceq1a ⊢ ( 𝑥 = 𝑦 → ( 𝜓 ↔ [ 𝑦 / 𝑥 ] 𝜓 ) )
10 7 8 9 3anbi123d ⊢ ( 𝑥 = 𝑦 → ( ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ∧ Tr 𝑦 ∧ [ 𝑦 / 𝑥 ] 𝜓 ) ) )
11 5 6 10 elabf ⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } ↔ ( [ 𝑦 / 𝑥 ] 𝜑 ∧ Tr 𝑦 ∧ [ 𝑦 / 𝑥 ] 𝜓 ) )
12 11 simp2bi ⊢ ( 𝑦 ∈ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) } → Tr 𝑦 )
13 1 12 mprg ⊢ Tr ∪ { 𝑥 ∣ ( 𝜑 ∧ Tr 𝑥 ∧ 𝜓 ) }