Theorem lemma-L6.03a 728

Description: Similar to solvedsub-P6a 711 but using ax-L12 29.

'𝑦' cannot occur in '𝑡'.

Hypotheses

Ref	Expression
lemma-L6.03a.1	⊢ Ⅎ𝑥𝜓
lemma-L6.03a.2	⊢ Ⅎ𝑦𝜒
lemma-L6.03a.3	⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
lemma-L6.03a.4	⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))

Assertion

Ref	Expression
lemma-L6.03a	⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ 𝜒)

Distinct variable groups:   𝑡,𝑦   𝑥,𝑦

Proof of Theorem lemma-L6.03a

Step	Hyp	Ref	Expression
1		lemma-L6.03a.1	. . . . 5 ⊢ Ⅎ𝑥𝜓
2		lemma-L6.03a.3	. . . . 5 ⊢ (𝑥 = 𝑦 → (𝜑 ↔ 𝜓))
3	1, 2	lemma-L6.02a 726	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ 𝜓)
4	3	subimr-P3.40b.RC 328	. . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → 𝜓))
5	4	suballinf-P5 594	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → 𝜓))
6		lemma-L6.03a.2	. . 3 ⊢ Ⅎ𝑦𝜒
7		lemma-L6.03a.4	. . 3 ⊢ (𝑦 = 𝑡 → (𝜓 ↔ 𝜒))
8	6, 7	lemma-L6.02a 726	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → 𝜓) ↔ 𝜒)
9	5, 8	bitrns-P3.33c.RC 303	1 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ 𝜒)

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104  Ⅎwff-nfree 681

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

This theorem is referenced by:  isubtopsub-P6  729