Theorem elabgft1 10588

Description: One implication of elabgf 2736, in closed form. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf1.nf1	⊢ Ⅎ𝑥𝐴
elabgf1.nf2	⊢ Ⅎ𝑥𝜓

Assertion

Ref	Expression
elabgft1	⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Proof of Theorem elabgft1

Step	Hyp	Ref	Expression
1		bi1 116	. . . . . 6 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜑))
2		imim2 54	. . . . . 6 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))
3	1, 2	syl5 32	. . . . 5 ⊢ ((𝜑 → 𝜓) → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))
4	3	imim2i 12	. . . 4 ⊢ ((𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))))
5	4	alimi 1384	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))))
6		elabgf1.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
7		nfab1 2221	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
8	6, 7	nfel 2227	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
9		elabgf1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
10	8, 9	nfim 1504	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
11		elabgf0 10587	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
12	6, 10, 11	bj-vtoclgft 10585	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))) → (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))
13	5, 12	syl 14	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)))
14	13	pm2.43d 49	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Syntax hints:   → wi 4   ↔ wb 103  ∀wal 1282   = wceq 1284  Ⅎwnf 1389   ∈ wcel 1433  {cab 2067  Ⅎwnfc 2206

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603

This theorem is referenced by:  elabgf1  10589