Theorem dfpsubv-P7 977

Description: dfpsubv-P6 717, Derived from Natural Deduction Rules (restriction on '𝑡'). †

This form can be used whenever '𝑥' does not occur in '𝑡'.

Assertion

Ref	Expression
dfpsubv-P7	⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Distinct variable group:   𝑡,𝑥

Proof of Theorem dfpsubv-P7

Step	Hyp	Ref	Expression
1		ndpsub3-P7.15 840	. . 3 ⊢ Ⅎ𝑥[𝑡 / 𝑥]𝜑
2		ndpsub2-P7.14 839	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ [𝑡 / 𝑥]𝜑))
3	2	ndbier-P3.15 180	. . . 4 ⊢ (𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 → 𝜑))
4	3	imcomm-P3.27.RC 266	. . 3 ⊢ ([𝑡 / 𝑥]𝜑 → (𝑥 = 𝑡 → 𝜑))
5	1, 4	alli-P7 947	. 2 ⊢ ([𝑡 / 𝑥]𝜑 → ∀𝑥(𝑥 = 𝑡 → 𝜑))
6	2	ndbief-P3.14 179	. . . . 5 ⊢ (𝑥 = 𝑡 → (𝜑 → [𝑡 / 𝑥]𝜑))
7	6	axL2-P3.22.RC 255	. . . 4 ⊢ ((𝑥 = 𝑡 → 𝜑) → (𝑥 = 𝑡 → [𝑡 / 𝑥]𝜑))
8	7	alloverim-P7.GENF.RC 972	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → ∀𝑥(𝑥 = 𝑡 → [𝑡 / 𝑥]𝜑))
9	1	qimeqallb-P7 976	. . . . 5 ⊢ (∀𝑥(𝑥 = 𝑡 → [𝑡 / 𝑥]𝜑) ↔ (∃𝑥 𝑥 = 𝑡 → ∀𝑥[𝑡 / 𝑥]𝜑))
10	9	rcp-NDBIEF0 240	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑡 → [𝑡 / 𝑥]𝜑) → (∃𝑥 𝑥 = 𝑡 → ∀𝑥[𝑡 / 𝑥]𝜑))
11		axL6ex-P7 925	. . . 4 ⊢ ∃𝑥 𝑥 = 𝑡
12	10, 11	mae-P3.23.RC 258	. . 3 ⊢ (∀𝑥(𝑥 = 𝑡 → [𝑡 / 𝑥]𝜑) → ∀𝑥[𝑡 / 𝑥]𝜑)
13		alle-P7.CL 942	. . 3 ⊢ (∀𝑥[𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜑)
14	8, 12, 13	dsyl-P3.25.RC 262	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) → [𝑡 / 𝑥]𝜑)
15	5, 14	rcp-NDBII0 239	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  ∃wff-exists 595  [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-L11 28  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:  dfpsub-P7  978  axL12-P7  982