Theorem nfabd 2509

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 451	. 2 ⊢ ((φ ∧ ¬ ∀x x = y) → Ⅎxψ)
4	1, 3	nfabd2 2508	1 ⊢ (φ → Ⅎx{y ∣ ψ})

Syntax hints:  ¬ wn 3   → wi 4  ∀wal 1540  Ⅎwnf 1544   = wceq 1642  {cab 2339  Ⅎwnfc 2477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479

This theorem is referenced by:  nfsbcd  3067  nfcsb1d  3167  nfcsbd  3170  nfifd  3686  nfunid  3899  nfiotad  4343