Theorem dfon2lem1 31688

Description: Lemma for dfon2 31697. (Contributed by Scott Fenton, 28-Feb-2011.)

Assertion

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

Proof of Theorem dfon2lem1

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		truni 4767	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)})
2		nfsbc1v 3455	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		nfv 1843	. . . . 5 ⊢ Ⅎ𝑥Tr 𝑦
4		nfsbc1v 3455	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
5	2, 3, 4	nf3an 1831	. . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)
6		vex 3203	. . . 4 ⊢ 𝑦 ∈ V
7		sbceq1a 3446	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		treq 4758	. . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9		sbceq1a 3446	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
10	7, 8, 9	3anbi123d 1399	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)))
11	5, 6, 10	elabf 3349	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))
12	11	simp2bi 1077	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦)
13	1, 12	mprg 2926	1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Syntax hints:   ∧ w3a 1037   ∈ wcel 1990  {cab 2608  [wsbc 3435  ∪ cuni 4436  Tr wtr 4752

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-sbc 3436  df-in 3581  df-ss 3588  df-uni 4437  df-iun 4522  df-tr 4753

This theorem is referenced by:  dfon2lem3  31690  dfon2lem7  31694