Theorem elabgf1 15392

Description: One implication of elabgf 2906. (Contributed by BJ, 21-Nov-2019.)

Hypotheses

Ref	Expression
elabgf1.nf1	⊢ Ⅎ𝑥𝐴
elabgf1.nf2	⊢ Ⅎ𝑥𝜓
elabgf1.1	⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))

Assertion

Ref	Expression
elabgf1	⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Proof of Theorem elabgf1

Step	Hyp	Ref	Expression
1		elabgf1.nf1	. . 3 ⊢ Ⅎ𝑥𝐴
2		elabgf1.nf2	. . 3 ⊢ Ⅎ𝑥𝜓
3	1, 2	elabgft1 15391	. 2 ⊢ (∀𝑥(𝑥 = 𝐴 → (𝜑 → 𝜓)) → (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓))
4		elabgf1.1	. 2 ⊢ (𝑥 = 𝐴 → (𝜑 → 𝜓))
5	3, 4	mpg 1465	1 ⊢ (𝐴 ∈ {𝑥 ∣ 𝜑} → 𝜓)

Syntax hints:   → wi 4   = wceq 1364  Ⅎwnf 1474   ∈ wcel 2167  {cab 2182  Ⅎwnfc 2326

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765

This theorem is referenced by:  elabf1  15394