Theorem 2stdpc4 2107

Description: A double specialization using explicit substitution. This is Theorem PM*11.1 in [WhiteheadRussell] p. 159. See stdpc4 2105 for the analogous single specialization. See 2sp 2225 for another double specialization. (Contributed by Andrew Salmon, 24-May-2011.)

Assertion

Ref	Expression
2stdpc4	⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

Proof of Theorem 2stdpc4

Step	Hyp	Ref	Expression
1		stdpc4 2105	. . 3 ⊢ (∀𝑦𝜑 → [𝑤 / 𝑦]𝜑)
2	1	alimi 1844	. 2 ⊢ (∀𝑥∀𝑦𝜑 → ∀𝑥[𝑤 / 𝑦]𝜑)
3		stdpc4 2105	. 2 ⊢ (∀𝑥[𝑤 / 𝑦]𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)
4	2, 3	syl 18	1 ⊢ (∀𝑥∀𝑦𝜑 → [𝑧 / 𝑥][𝑤 / 𝑦]𝜑)

This proof depends on syntax axioms:   → wi 4  ∀wal 1568  [wsb 2099

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100

This theorem is used by:  ax11-pm2  37512  pm11.11  45125