Theorem elabgft1 13813

Description: One implication of elabgf 2872, 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		biimp 117	. . . . . 6 ⊢ ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜑))
2		imim2 55	. . . . . 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 1448	. . 3 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → ∀𝑥(𝑥 = 𝐴 → ((𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))))
6		elabgf1.nf1	. . . 4 ⊢ Ⅎ𝑥𝐴
7		nfab1 2314	. . . . . 6 ⊢ Ⅎ𝑥{𝑥 ∣ 𝜑}
8	6, 7	nfel 2321	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ {𝑥 ∣ 𝜑}
9		elabgf1.nf2	. . . . 5 ⊢ Ⅎ𝑥𝜓
10	8, 9	nfim 1565	. . . 4 ⊢ Ⅎ𝑥(𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)
11		elabgf0 13812	. . . 4 ⊢ (𝑥 = 𝐴 → (𝐴 ∈ {𝑥 ∣ 𝜑} ↔ 𝜑))
12	6, 10, 11	bj-vtoclgft 13810	. . 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 50	1 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))

Syntax hints:   → wi 4   ↔ wb 104  ∀wal 1346   = wceq 1348  Ⅎwnf 1453   ∈ wcel 2141  {cab 2156  Ⅎwnfc 2299

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732

This theorem is referenced by:  elabgf1  13814