Theorem nfrsucc-P6 780

Description: If '𝑥' is ENF a term '𝑡', then '𝑥' is also ENF in its successor 's‘𝑡'.

'𝑎' is distinct from all other variables.

Hypothesis

Ref	Expression
nfrsucc-P6.1	⊢ Ⅎ𝑥 𝑎 = 𝑡

Assertion

Ref	Expression
nfrsucc-P6	⊢ Ⅎ𝑥 𝑎 = s‘𝑡

Distinct variable groups:   𝑡,𝑎   𝑥,𝑎

Proof of Theorem nfrsucc-P6

Dummy variable 𝑏 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfrv-P6 686	. . . 4 ⊢ Ⅎ𝑥 𝑎 = s‘𝑏
2	1	rcp-NDIMP0addall 207	. . 3 ⊢ (𝑏 = 𝑡 → Ⅎ𝑥 𝑎 = s‘𝑏)
3		nfrsucc-P6.1	. . . . 5 ⊢ Ⅎ𝑥 𝑎 = 𝑡
4	3	nfrterm-P6 779	. . . 4 ⊢ Ⅎ𝑥 𝑏 = 𝑡
5		ax-L9-succ 22	. . . . 5 ⊢ (𝑏 = 𝑡 → s‘𝑏 = s‘𝑡)
6	5	subeqr-P5 635	. . . 4 ⊢ (𝑏 = 𝑡 → (𝑎 = s‘𝑏 ↔ 𝑎 = s‘𝑡))
7	4, 6	subnfr-P6 755	. . 3 ⊢ (𝑏 = 𝑡 → (Ⅎ𝑥 𝑎 = s‘𝑏 ↔ Ⅎ𝑥 𝑎 = s‘𝑡))
8	2, 7	bimpf-P4 531	. 2 ⊢ (𝑏 = 𝑡 → Ⅎ𝑥 𝑎 = s‘𝑡)
9		axL6ex-P5 625	. 2 ⊢ ∃𝑏 𝑏 = 𝑡
10	8, 9	exiav-P5.SH 616	1 ⊢ Ⅎ𝑥 𝑎 = s‘𝑡

Syntax hints:  term-obj 1  s‘term_succ 3   = wff-equals 6  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  ax-L9-succ 22  ax-L12 29

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:  psubsuccv-P6  806