Theorem nfrterm-P6 779

Description: Changing the dummy variable doesn't change the ENF state.

'𝑎' and '𝑏' are distinct from all other variables.

Hypothesis

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

Assertion

Ref	Expression
nfrterm-P6	⊢ Ⅎ𝑥 𝑏 = 𝑡

Distinct variable groups:   𝑡,𝑎,𝑏   𝑥,𝑎,𝑏

Proof of Theorem nfrterm-P6

Step	Hyp	Ref	Expression
1		nfrterm-P6.1	. . . 4 ⊢ Ⅎ𝑥 𝑎 = 𝑡
2	1	ax-GEN 15	. . 3 ⊢ ∀𝑎Ⅎ𝑥 𝑎 = 𝑡
3		subeql-P5.CL 633	. . . . 5 ⊢ (𝑎 = 𝑏 → (𝑎 = 𝑡 ↔ 𝑏 = 𝑡))
4	3	subnfr-P6.VR 756	. . . 4 ⊢ (𝑎 = 𝑏 → (Ⅎ𝑥 𝑎 = 𝑡 ↔ Ⅎ𝑥 𝑏 = 𝑡))
5	4	cbvallv-P5 659	. . 3 ⊢ (∀𝑎Ⅎ𝑥 𝑎 = 𝑡 ↔ ∀𝑏Ⅎ𝑥 𝑏 = 𝑡)
6	2, 5	bimpf-P4.RC 532	. 2 ⊢ ∀𝑏Ⅎ𝑥 𝑏 = 𝑡
7		spec-P6 719	. 2 ⊢ (∀𝑏Ⅎ𝑥 𝑏 = 𝑡 → Ⅎ𝑥 𝑏 = 𝑡)
8	6, 7	rcp-NDIME0 228	1 ⊢ Ⅎ𝑥 𝑏 = 𝑡

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8  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-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:  nfrsucc-P6  780  nfradd-P6  781  nfrmult-P6  782  psubsuccv-P6  806  psubaddv-P6  808  psubmultv-P6  810