Theorem nfabd 2785

Description: Bound-variable hypothesis builder for a class abstraction. (Contributed by Mario Carneiro, 8-Oct-2016.)

Hypotheses

Ref	Expression
nfabd.1	⊢ Ⅎ𝑦𝜑
nfabd.2	⊢ (𝜑 → Ⅎ𝑥𝜓)

Assertion

Ref	Expression
nfabd	⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Proof of Theorem nfabd

Step	Hyp	Ref	Expression
1		nfabd.1	. 2 ⊢ Ⅎ𝑦𝜑
2		nfabd.2	. . 3 ⊢ (𝜑 → Ⅎ𝑥𝜓)
3	2	adantr 481	. 2 ⊢ ((𝜑 ∧ ¬ ∀𝑥 𝑥 = 𝑦) → Ⅎ𝑥𝜓)
4	1, 3	nfabd2 2784	1 ⊢ (𝜑 → Ⅎ𝑥{𝑦 ∣ 𝜓})

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1481  Ⅎwnf 1708  {cab 2608  Ⅎwnfc 2751

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753

This theorem is referenced by:  nfsbcd  3456  nfcsb1d  3547  nfcsbd  3550  nfifd  4114  nfunid  4443  nfiotad  5854  nfintd  42420  nfiund  42421