Theorem nfrexdya 2541

Description: Not-free for restricted existential quantification where 𝑦 and 𝐴 are distinct. See nfrexdxy 2539 for a version with 𝑥 and 𝑦 distinct instead. (Contributed by Jim Kingdon, 30-May-2018.)

Hypotheses

Ref	Expression
nfraldya.2	⊢ Ⅎ𝑦𝜑
nfraldya.3	⊢ (𝜑 → Ⅎ𝑥𝐴)
nfraldya.4	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfrexdya	⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Distinct variable group:   𝑦,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥,𝑦)   𝐴(𝑥)

Proof of Theorem nfrexdya

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2489	. 2 ⊢ (∃𝑦 ∈ 𝐴 𝜓 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
2		sban 1982	. . . . . 6 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓))
3		clelsb1 2309	. . . . . . 7 ⊢ ([𝑧 / 𝑦]𝑦 ∈ 𝐴 ↔ 𝑧 ∈ 𝐴)
4	3	anbi1i 458	. . . . . 6 ⊢ (([𝑧 / 𝑦]𝑦 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓))
5	2, 4	bitri 184	. . . . 5 ⊢ ([𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ (𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓))
6	5	exbii 1627	. . . 4 ⊢ (∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓))
7		nfv 1550	. . . . 5 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐴 ∧ 𝜓)
8	7	sb8e 1879	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧[𝑧 / 𝑦](𝑦 ∈ 𝐴 ∧ 𝜓))
9		df-rex 2489	. . . 4 ⊢ (∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓 ↔ ∃𝑧(𝑧 ∈ 𝐴 ∧ [𝑧 / 𝑦]𝜓))
10	6, 8, 9	3bitr4i 212	. . 3 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓) ↔ ∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
11		nfv 1550	. . . 4 ⊢ Ⅎ𝑧𝜑
12		nfraldya.3	. . . 4 ⊢ (𝜑 → Ⅎ𝑥𝐴)
13		nfraldya.2	. . . . 5 ⊢ Ⅎ𝑦𝜑
14		nfraldya.4	. . . . 5 ⊢ (𝜑 → Ⅎ𝑥𝜓)
15	13, 14	nfsbd 2004	. . . 4 ⊢ (𝜑 → Ⅎ𝑥[𝑧 / 𝑦]𝜓)
16	11, 12, 15	nfrexdxy 2539	. . 3 ⊢ (𝜑 → Ⅎ𝑥∃𝑧 ∈ 𝐴 [𝑧 / 𝑦]𝜓)
17	10, 16	nfxfrd 1497	. 2 ⊢ (𝜑 → Ⅎ𝑥∃𝑦(𝑦 ∈ 𝐴 ∧ 𝜓))
18	1, 17	nfxfrd 1497	1 ⊢ (𝜑 → Ⅎ𝑥∃𝑦 ∈ 𝐴 𝜓)

Syntax hints:   → wi 4   ∧ wa 104  Ⅎwnf 1482  ∃wex 1514  [wsb 1784   ∈ wcel 2175  Ⅎwnfc 2334  ∃wrex 2484

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-nf 1483  df-sb 1785  df-cleq 2197  df-clel 2200  df-nfc 2336  df-rex 2489

This theorem is referenced by:  nfrexya  2546