Theorem nfrv-P6 686

Description: Does Not Occur in ⇒ ENF in.

If '𝑥' does not occur in '𝜑', then '𝑥' is effectively not free in '𝜑'.

Assertion

Ref	Expression
nfrv-P6	⊢ Ⅎ𝑥𝜑

Distinct variable group:   𝜑,𝑥

Proof of Theorem nfrv-P6

Dummy variable 𝑥₁ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biref-P3.33a 297	. . 3 ⊢ (𝜑 ↔ 𝜑)
2	1	rcp-NDIMP0addall 207	. 2 ⊢ (𝑥 = 𝑥₁ → (𝜑 ↔ 𝜑))
3		ax-L5 17	. 2 ⊢ (𝜑 → ∀𝑥𝜑)
4	2, 3	gennfrw-P6 685	1 ⊢ Ⅎ𝑥𝜑

Syntax hints:  term-obj 1   = wff-equals 6   ↔ wff-bi 104  Ⅎwff-nfree 681

This theorem was proved from axioms:  ax-L1 11  ax-L2 12  ax-L3 13  ax-MP 14  ax-GEN 15  ax-L4 16  ax-L5 17  ax-L6 18  ax-L7 19

This theorem depends on definitions:  df-bi-D2.1 107  df-and-D2.2 133  df-or-D2.3 145  df-true-D2.4 155  df-rcp-AND3 161  df-exists-D5.1 596  df-nfree-D6.1 682

This theorem is referenced by:  solvesub-P6a.VR  705  example-E6.01a  706  trnsvsubw-P6  710  example-E6.02a  712  subnfr-P6.VR  756  psubtoisub-P6  765  spliteq-P6  776  splitelof-P6  778  nfrsucc-P6  780  nfradd-P6  781  nfrmult-P6  782  psubnfr-P6.VR  785  psubvar1-P6  802  ndnfrv-P7.1  826