Theorem psuball2v-P6 796

Description: Proper Substitution Over Universal Quantifier (different variable - restriction on '𝑡').

'𝑦' cannot occur in '𝑡'.

Assertion

Ref	Expression
psuball2v-P6	⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝑡 / 𝑥]𝜑)

Distinct variable groups:   𝑡,𝑦   𝑥,𝑦

Proof of Theorem psuball2v-P6

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		psubcomp-P6 767	. 2 ⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ [𝑡 / 𝑧][𝑧 / 𝑥]∀𝑦𝜑)
2		psuball2v-P6-L1 795	. . 3 ⊢ ([𝑧 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝑧 / 𝑥]𝜑)
3	2	psubleq-P6 783	. 2 ⊢ ([𝑡 / 𝑧][𝑧 / 𝑥]∀𝑦𝜑 ↔ [𝑡 / 𝑧]∀𝑦[𝑧 / 𝑥]𝜑)
4		psuball2v-P6-L1 795	. 2 ⊢ ([𝑡 / 𝑧]∀𝑦[𝑧 / 𝑥]𝜑 ↔ ∀𝑦[𝑡 / 𝑧][𝑧 / 𝑥]𝜑)
5		psubcomp-P6 767	. . . 4 ⊢ ([𝑡 / 𝑥]𝜑 ↔ [𝑡 / 𝑧][𝑧 / 𝑥]𝜑)
6	5	bisym-P3.33b.RC 299	. . 3 ⊢ ([𝑡 / 𝑧][𝑧 / 𝑥]𝜑 ↔ [𝑡 / 𝑥]𝜑)
7	6	suballinf-P5 594	. 2 ⊢ (∀𝑦[𝑡 / 𝑧][𝑧 / 𝑥]𝜑 ↔ ∀𝑦[𝑡 / 𝑥]𝜑)
8	1, 3, 4, 7	tbitrns-P4.17.RC 431	1 ⊢ ([𝑡 / 𝑥]∀𝑦𝜑 ↔ ∀𝑦[𝑡 / 𝑥]𝜑)

Syntax hints:  term-obj 1  ∀wff-forall 8   ↔ wff-bi 104  [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:  psubex2v-P6  797  psuball2-P6  798