Theorem keephyp 2392

Description: Transform a hypothesis ψ that we want to keep (but contains the same class variable A used in the eliminated hypothesis) for use with the weak deduction theorem.

Hypotheses

Ref	Expression
keephyp.1	⊢ (A = if(φ, A, B) → (ψ ↔ θ))
keephyp.2	⊢ (B = if(φ, A, B) → (χ ↔ θ))
keephyp.3	⊢ ψ
keephyp.4	⊢ χ

Assertion

Ref	Expression
keephyp	⊢ θ

Proof of Theorem keephyp

Step	Hyp	Ref	Expression
1		keephyp.3	. 2 ⊢ ψ
2		keephyp.4	. 2 ⊢ χ
3		keephyp.1	. . 3 ⊢ (A = if(φ, A, B) → (ψ ↔ θ))
4		keephyp.2	. . 3 ⊢ (B = if(φ, A, B) → (χ ↔ θ))
5	3, 4	ifboth 2371	. 2 ⊢ ((ψ ⋀ χ) → θ)
6	1, 2, 5	mp2an 696	1 ⊢ θ

Syntax hints:   → wi 3   ↔ wb 146   = wceq 954   ifcif 2357

This theorem is referenced by:  keepel 2395  mulcant2 5668  sqrlem21 6631  sqrlem22 6632  projlem7 9131

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-if 2358

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