Theorem nfcrALT 2890

Description: Alternate version of nfcr 2889. Avoids ax-8 2116 but uses ax-12 2185. (Contributed by Mario Carneiro, 11-Aug-2016.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
nfcrALT	⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Distinct variable groups:   𝑥,𝑦   𝑦,𝐴

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem nfcrALT

Step	Hyp	Ref	Expression
1		df-nfc 2886	. 2 ⊢ (Ⅎ𝑥𝐴 ↔ ∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴)
2		sp 2191	. 2 ⊢ (∀𝑦Ⅎ𝑥 𝑦 ∈ 𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)
3	1, 2	sylbi 217	1 ⊢ (Ⅎ𝑥𝐴 → Ⅎ𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1540  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-12 2185

This theorem depends on definitions:  df-bi 207  df-ex 1782  df-nfc 2886

This theorem is referenced by: (None)