Theorem psubjust-P6 715

Description: Justification Theorem for df-psub-D6.2 716.

'𝑦' and '𝑧' are distinct from all other variables.

Assertion

Ref	Expression
psubjust-P6	⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑦 = 𝑥 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑧 = 𝑥 → 𝜑)))

Distinct variable groups:   𝜑,𝑦,𝑧   𝑡,𝑦,𝑧   𝑥,𝑦,𝑧

Proof of Theorem psubjust-P6

Step	Hyp	Ref	Expression
1		subeql-P5.CL 633	. . 3 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑡 ↔ 𝑧 = 𝑡))
2		subeql-P5.CL 633	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 = 𝑥 ↔ 𝑧 = 𝑥))
3	2	subiml-P3.40a 325	. . . 4 ⊢ (𝑦 = 𝑧 → ((𝑦 = 𝑥 → 𝜑) ↔ (𝑧 = 𝑥 → 𝜑)))
4	3	suballv-P5 623	. . 3 ⊢ (𝑦 = 𝑧 → (∀𝑥(𝑦 = 𝑥 → 𝜑) ↔ ∀𝑥(𝑧 = 𝑥 → 𝜑)))
5	1, 4	subimd-P3.40c 329	. 2 ⊢ (𝑦 = 𝑧 → ((𝑦 = 𝑡 → ∀𝑥(𝑦 = 𝑥 → 𝜑)) ↔ (𝑧 = 𝑡 → ∀𝑥(𝑧 = 𝑥 → 𝜑))))
6	5	cbvallv-P5 659	1 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑦 = 𝑥 → 𝜑)) ↔ ∀𝑧(𝑧 = 𝑡 → ∀𝑥(𝑧 = 𝑥 → 𝜑)))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104

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

This theorem is referenced by: (None)