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 ) }