Theorem lemma-L6.08a 773

Description: Equivalency of definitions lemma.

'𝑦' cannot occur in '𝑡'.

Assertion

Ref	Expression
lemma-L6.08a	⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Distinct variable groups:   𝑡,𝑦   𝑥,𝑦

Proof of Theorem lemma-L6.08a

Step	Hyp	Ref	Expression
1		lemma-L6.07a 772	. . . 4 ⊢ (∀𝑥(𝑥 = 𝑦 → 𝜑) ↔ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑))
2	1	subimr-P3.40b.RC 328	. . 3 ⊢ ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ (𝑦 = 𝑡 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
3	2	suballinf-P5 594	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
4		lemma-L6.07a 772	. 2 ⊢ (∀𝑦(𝑦 = 𝑡 → ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))
5	3, 4	bitrns-P3.33c.RC 303	1 ⊢ (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦 → 𝜑)) ↔ ∃𝑦(𝑦 = 𝑡 ∧ ∃𝑥(𝑥 = 𝑦 ∧ 𝜑)))

Syntax hints:  term-obj 1   = wff-equals 6  ∀wff-forall 8   → wff-imp 10   ↔ wff-bi 104   ∧ wff-and 132  ∃wff-exists 595

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-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:  dfpsubalt-P6  774