Theorem lemma-L6.06a 766

Description: Effective Non-Freeness Over Proper Substitution (restriction on '𝑡').

Note this only holds when '𝑡' does not contain '𝑥'.

Assertion

Ref	Expression
lemma-L6.06a	⊢ Ⅎ𝑥[𝑡 / 𝑥]𝜑

Distinct variable group:   𝑡,𝑥

Proof of Theorem lemma-L6.06a

Step	Hyp	Ref	Expression
1		nfrall1-P6 741	. 2 ⊢ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑)
2		dfpsubv-P6 717	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
3	2	nfrleq-P6 687	. 2 ⊢ (Ⅎ𝑥[𝑡 / 𝑥]𝜑 ↔ Ⅎ𝑥∀𝑥(𝑥 = 𝑡 → 𝜑))
4	1, 3	bimpr-P4.RC 534	1 ⊢ Ⅎ𝑥[𝑡 / 𝑥]𝜑

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10  Ⅎwff-nfree 681  [wff-psub 714

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-L10 27  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  df-psub-D6.2 716

This theorem is referenced by:  psubcomp-P6  767  cbvallpsub-P6  768  cbvexpsub-P6  769  psuball2v-P6-L1  795  psubsuccv-P6  806  psubaddv-P6  808  psubmultv-P6  810  ndpsub3-P7.15  840