Theorem psubspliteq-P6 800

Description: Apply Proper Substitution to Split Equlity.

'𝑎' is distinct from all other variables.

Assertion

Ref	Expression
psubspliteq-P6	⊢ ([𝑤 / 𝑥] 𝑡 = 𝑢 ↔ ∃𝑎([𝑤 / 𝑥] 𝑎 = 𝑡 ∧ [𝑤 / 𝑥] 𝑎 = 𝑢))

Distinct variable groups:   𝑡,𝑎   𝑢,𝑎   𝑤,𝑎   𝑥,𝑎

Proof of Theorem psubspliteq-P6

Step	Hyp	Ref	Expression
1		spliteq-P6 776	. . 3 ⊢ (𝑡 = 𝑢 ↔ ∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
2	1	psubleq-P6 783	. 2 ⊢ ([𝑤 / 𝑥] 𝑡 = 𝑢 ↔ [𝑤 / 𝑥]∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
3		psubex2v-P6 797	. 2 ⊢ ([𝑤 / 𝑥]∃𝑎(𝑎 = 𝑡 ∧ 𝑎 = 𝑢) ↔ ∃𝑎[𝑤 / 𝑥](𝑎 = 𝑡 ∧ 𝑎 = 𝑢))
4		psuband-P6 792	. . 3 ⊢ ([𝑤 / 𝑥](𝑎 = 𝑡 ∧ 𝑎 = 𝑢) ↔ ([𝑤 / 𝑥] 𝑎 = 𝑡 ∧ [𝑤 / 𝑥] 𝑎 = 𝑢))
5	4	subexinf-P5 608	. 2 ⊢ (∃𝑎[𝑤 / 𝑥](𝑎 = 𝑡 ∧ 𝑎 = 𝑢) ↔ ∃𝑎([𝑤 / 𝑥] 𝑎 = 𝑡 ∧ [𝑤 / 𝑥] 𝑎 = 𝑢))
6	2, 3, 5	dbitrns-P4.16.RC 429	1 ⊢ ([𝑤 / 𝑥] 𝑡 = 𝑢 ↔ ∃𝑎([𝑤 / 𝑥] 𝑎 = 𝑡 ∧ [𝑤 / 𝑥] 𝑎 = 𝑢))

Syntax hints:  term-obj 1   = wff-equals 6   ↔ wff-bi 104   ∧ wff-and 132  ∃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: (None)