Theorem isubtopsub-P6 729

Description: Conversion from Implicit to Explicit Substitution with Intermediate Formula.

'𝑦' cannot occur in either '𝜑' or '𝑡'.

If '𝑦' is a fresh variable that appears in the intermediate WFF '𝜓', then we can substitute '𝑡' for '𝑥' even when '𝑥' occurs in '𝑡'.

Hypotheses

Ref	Expression
isubtopsub-P6.1	⊢ Ⅎ𝑥𝜓
isubtopsub-P6.2	⊢ Ⅎ𝑦𝜒
isubtopsub-P6.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
isubtopsub-P6.4	⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
isubtopsub-P6	⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜒)

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

Proof of Theorem isubtopsub-P6

Step	Hyp	Ref	Expression
1		df-psub-D6.2 716	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)))
2		isubtopsub-P6.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		isubtopsub-P6.2	. . 3 ⊢ Ⅎ𝑦𝜒
4		isubtopsub-P6.3	. . 3 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
5		isubtopsub-P6.4	. . 3 ⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))
6	2, 3, 4, 5	lemma-L6.03a 728	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ 𝜒)
7	1, 6	bitrns-P3.33c.RC 303	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜒)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ 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-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:  psubcomp-P6  767  psubvar1-P6  802