Theorem nfcrii 2972

Description: Consequence of the not-free predicate. (Contributed by Mario Carneiro, 11-Aug-2016.)

Hypothesis

Ref	Expression
nfcri.1	⊢ Ⅎ𝑥𝐴

Assertion

Ref	Expression
nfcrii	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Distinct variable group:   𝑥,𝑦

Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem nfcrii

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcri.1	. . . . 5 ⊢ Ⅎ𝑥𝐴
2	1	nfcriv 2969	. . . 4 ⊢ Ⅎ𝑥 𝑧 ∈ 𝐴
3	2	nfsbv 2349	. . 3 ⊢ Ⅎ𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴
4	3	nf5ri 2195	. 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 → ∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴)
5		clelsb3 2942	. 2 ⊢ ([𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
6	5	albii 1820	. 2 ⊢ (∀𝑥[𝑦 / 𝑧]𝑧 ∈ 𝐴 ↔ ∀𝑥 𝑦 ∈ 𝐴)
7	4, 5, 6	3imtr3i 293	1 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Syntax hints:   → wi 4  ∀wal 1535  [wsb 2069   ∈ wcel 2114  Ⅎwnfc 2963

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-10 2145  ax-11 2161  ax-12 2177

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-nf 1785  df-sb 2070  df-clel 2895  df-nfc 2965

This theorem is referenced by:  nfcri  2973  cleqfOLD  3013  abeq2fOLD  3016  bnj1230  32076  bnj1000  32215  bnj1204  32286  bnj1307  32297  bnj1311  32298  bnj1398  32308  bnj1466  32327  bnj1467  32328  bnj1523  32345