Theorem psuball2-P6 798

Description: Proper Substitution Over Universal Quantifier (different variable - non-freeness condition).

'𝑧' cannot occur in '𝑡'.

Hypothesis

Ref	Expression
psuball2-P6.1	⊢ Ⅎ𝑧𝜑

Assertion

Ref	Expression
psuball2-P6	⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ ∀𝑧[𝑡 / 𝑥][𝑧 / 𝑦]𝜑)

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

Proof of Theorem psuball2-P6

Step	Hyp	Ref	Expression
1		psuball2-P6.1	. . . 4 ⊢ Ⅎ𝑧𝜑
2	1	cbvallpsub-P6 768	. . 3 ⊢ (∀𝑦𝜑 ↔ ∀𝑧[𝑧 / 𝑦]𝜑)
3	2	psubleq-P6 783	. 2 ⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ [𝑡 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑)
4		psuball2v-P6 796	. 2 ⊢ ([𝑡 / 𝑥]∀𝑧[𝑧 / 𝑦]𝜑 ↔ ∀𝑧[𝑡 / 𝑥][𝑧 / 𝑦]𝜑)
5	3, 4	bitrns-P3.33c.RC 303	1 ⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ ∀𝑧[𝑡 / 𝑥][𝑧 / 𝑦]𝜑)

Syntax hints:  term-obj 1  ∀wff-forall 8   ↔ wff-bi 104  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-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: (None)