Theorem psubex2v-P6 797

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

'𝑦' cannot occur in '𝑡'.

Assertion

Ref	Expression
psubex2v-P6	⊢ ([𝑡 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝑡 / 𝑥]𝜑)

Distinct variable groups:   𝑡,𝑦   𝑥,𝑦

Proof of Theorem psubex2v-P6

Step	Hyp	Ref	Expression
1		df-exists-D5.1 596	. . 3 ⊢ (∃𝑦𝜑 ↔ ¬ ∀𝑦 ¬ 𝜑)
2	1	psubleq-P6 783	. 2 ⊢ ([𝑡 / 𝑥]∃𝑦𝜑 ↔ [𝑡 / 𝑥] ¬ ∀𝑦 ¬ 𝜑)
3		psubneg-P6 788	. 2 ⊢ ([𝑡 / 𝑥] ¬ ∀𝑦 ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]∀𝑦 ¬ 𝜑)
4		psuball2v-P6 796	. . . 4 ⊢ ([𝑡 / 𝑥]∀𝑦 ¬ 𝜑 ↔ ∀𝑦[𝑡 / 𝑥] ¬ 𝜑)
5		psubneg-P6 788	. . . . 5 ⊢ ([𝑡 / 𝑥] ¬ 𝜑 ↔ ¬ [𝑡 / 𝑥]𝜑)
6	5	suballinf-P5 594	. . . 4 ⊢ (∀𝑦[𝑡 / 𝑥] ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑡 / 𝑥]𝜑)
7	4, 6	bitrns-P3.33c.RC 303	. . 3 ⊢ ([𝑡 / 𝑥]∀𝑦 ¬ 𝜑 ↔ ∀𝑦 ¬ [𝑡 / 𝑥]𝜑)
8	7	subneg-P3.39.RC 324	. 2 ⊢ (¬ [𝑡 / 𝑥]∀𝑦 ¬ 𝜑 ↔ ¬ ∀𝑦 ¬ [𝑡 / 𝑥]𝜑)
9		df-exists-D5.1 596	. . 3 ⊢ (∃𝑦[𝑡 / 𝑥]𝜑 ↔ ¬ ∀𝑦 ¬ [𝑡 / 𝑥]𝜑)
10	9	bisym-P3.33b.RC 299	. 2 ⊢ (¬ ∀𝑦 ¬ [𝑡 / 𝑥]𝜑 ↔ ∃𝑦[𝑡 / 𝑥]𝜑)
11	2, 3, 8, 10	tbitrns-P4.17.RC 431	1 ⊢ ([𝑡 / 𝑥]∃𝑦𝜑 ↔ ∃𝑦[𝑡 / 𝑥]𝜑)

Syntax hints:  ∀wff-forall 8  ¬ wff-neg 9   ↔ 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:  psubex2-P6  799  psubspliteq-P6  800  psubsplitelof-P6  801