Theorem isubtopsubv-P6 727

Description: Conversion from Implicit to Explicit Substitution (restriction on '𝑡').

'𝑥' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
isubtopsubv-P6.1	⊢ Ⅎ𝑥𝜓
isubtopsubv-P6.2	⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))

Assertion

Ref	Expression
isubtopsubv-P6	⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜓)

Distinct variable group:   𝑡,𝑥

Proof of Theorem isubtopsubv-P6

Step	Hyp	Ref	Expression
1		dfpsubv-P6 717	. 2 ⊢ ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡 → 𝜑))
2		isubtopsubv-P6.1	. . 3 ⊢ Ⅎ𝑥𝜓
3		isubtopsubv-P6.2	. . 3 ⊢ (𝑥 = 𝑡 → (𝜑 ↔ 𝜓))
4	2, 3	lemma-L6.02a 726	. 2 ⊢ (∀𝑥(𝑥 = 𝑡 → 𝜑) ↔ 𝜓)
5	1, 4	bitrns-P3.33c.RC 303	1 ⊢ ([𝑡 / 𝑥]𝜑 ↔ 𝜓)

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:  psuball2v-P6-L1  795  psubsuccv-P6  806  psubaddv-P6  808  psubmultv-P6  810  ndpsub1-P7.13  838