Theorem psubim-P6 791

Description: Proper Substitution Over Implication.

Assertion

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

Proof of Theorem psubim-P6

Step	Hyp	Ref	Expression
1		psubim-P6-L1 789	. 2 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) → ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))
2		psubim-P6-L2 790	. 2 ⊢ (([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓) → [𝑡 / 𝑥](𝜑 → 𝜓))
3	1, 2	rcp-NDBII0 239	1 ⊢ ([𝑡 / 𝑥](𝜑 → 𝜓) ↔ ([𝑡 / 𝑥]𝜑 → [𝑡 / 𝑥]𝜓))

Syntax hints:   → wff-imp 10   ↔ wff-bi 104  [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-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:  psuband-P6  792