Theorem dfon2lem1 33035

Description: Lemma for dfon2 33044. (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 5172	. 2 ⊢ (∀𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}Tr 𝑦 → Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)})
2		nfsbc1v 3783	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜑
3		nfv 1915	. . . . 5 ⊢ Ⅎ𝑥Tr 𝑦
4		nfsbc1v 3783	. . . . 5 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜓
5	2, 3, 4	nf3an 1902	. . . 4 ⊢ Ⅎ𝑥([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)
6		vex 3489	. . . 4 ⊢ 𝑦 ∈ V
7		sbceq1a 3774	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ [𝑦 / 𝑥]𝜑))
8		treq 5164	. . . . 5 ⊢ (𝑥 = 𝑦 → (Tr 𝑥 ↔ Tr 𝑦))
9		sbceq1a 3774	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜓 ↔ [𝑦 / 𝑥]𝜓))
10	7, 8, 9	3anbi123d 1432	. . . 4 ⊢ (𝑥 = 𝑦 → ((𝜑 ∧ Tr 𝑥 ∧ 𝜓) ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓)))
11	5, 6, 10	elabf 3656	. . 3 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} ↔ ([𝑦 / 𝑥]𝜑 ∧ Tr 𝑦 ∧ [𝑦 / 𝑥]𝜓))
12	11	simp2bi 1142	. 2 ⊢ (𝑦 ∈ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)} → Tr 𝑦)
13	1, 12	mprg 3152	1 ⊢ Tr ∪ {𝑥 ∣ (𝜑 ∧ Tr 𝑥 ∧ 𝜓)}

Syntax hints:   ∧ w3a 1083   ∈ wcel 2114  {cab 2799  [wsbc 3763  ∪ cuni 4824  Tr wtr 5158

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3488  df-sbc 3764  df-in 3931  df-ss 3940  df-uni 4825  df-iun 4907  df-tr 5159

This theorem is referenced by:  dfon2lem3  33037  dfon2lem7  33041