Theorem elab1 13818

Description: One implication of elab 2874. (Contributed by BJ, 21-Nov-2019.)

Hypothesis

Ref	Expression
elab1.1	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
elab1	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Distinct variable groups:   𝜓,𝑥   𝑥,𝐴

Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elab1

Step	Hyp	Ref	Expression
1		nfv 1521	. 2 ⊢ Ⅎ𝑥𝜓
2		elab1.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
3	1, 2	elabf1 13816	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   = wceq 1348   ∈ wcel 2141  {cab 2156

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: (None)